Answer:
When a block with mass M moving at a velocity of V collides and sticks to a block of mass 2M initially at rest, their velocity after the collision is equal to the original velocity of the first block. This is because, in a collision where the two objects stick together, the final velocity of the combined system is equal to the initial velocity of the first object.
To calculate the final velocity of the combined system, we can use the formula vf = vi + (2mv)/(m1+m2), where vf is the final velocity, vi is the initial velocity, m is the mass of the first object, v is the velocity of the first object, and m1 and m2 are the masses of the two objects. Plugging in the values from the problem, we get:
vf = V + (2 * M * V)/(M + 2M) = V + (2 * M * V)/(3M) = V + (2/3) * V = (5/3) * V
Therefore, the final velocity of the combined system is (5/3) * V, which is equal to the original velocity of the first block.
The velocity of the mass 2M after the collision will be V/2 which is half of the velocity of mass M.
What is elastic collision?An elastic collision is one during which the system does not experience a huge reduction of kinetic energy as a result of the collision. In elastic collisions, mass and kinetic energy are both preserved. Imagine two trolleys that resemble one another moving in the same direction at the same pace.
A block with mass M moving at a velocity of V collides and sticks to a block of mass 2M initially at rest.
Let 'v' be the velocity after the collision. Then the velocity of the mass 2M after the collision will be calculated as,
MV = 2Mv
v = V / 2
Thus, the velocity of the mass 2M after the collision will be V/2 which is half of the velocity of the mass M.
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Which of the following research situations would be most likely to use an independent-measures design?
a) examine the development of vocabulary as a group of children mature from age 2 to age 3
b) examine the long-term effectiveness of a stop-smoking treatment by interviewing subjects 2 months and 7 months after the treatment ends
c) compare the mathematics skills for 9th grade boys versus 9th grade girls*
d) compare the blood-pressure readings before medication and after medication for a group a patients with high blood pressure
Option C, The research situation that would be most likely to use an independent-measures design is to compare the mathematics skills of 9th-grade boys versus 9th-grade girls.
An independent-measures design is used when there are two or more groups that are being compared and each individual only belongs to one group. In this research situation, the two groups being compared are 9th-grade boys and 9th-grade girls. Each individual in the study belongs to one group or the other, and they are not part of both groups. This makes it an ideal situation for using an independent-measures design.
Option a) examine the development of vocabulary as a group of children mature from age 2 to age 3 would likely use a within-subjects design, as the same group of children is being measured at two different time points.
Option b) Examining the long-term effectiveness of a stop-smoking treatment by interviewing subjects 2 months and 7 months after the treatment ends would also likely use a within-subjects design, as the same group of subjects is being measured at two different time points.
Option d) compare the blood-pressure readings before medication and after medication for a group of patients with high blood pressure would likely use a repeated-measures design, as the same group of patients is being measured at two different time points and the order of the measurements (before or after medication) is being controlled.
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Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
integral.gif C (3y + 7e^sqrt(x)) dx + (8x + 5 cos y^2) dy
C is the boundary of the region enclosed by the parabolas y = x2 and x = y2
On solving the provided question, From the question, our region is defined by: lower bound: y= [tex]x^2[/tex] and upper bound: y = [tex]\sqrt{x}[/tex]
what is integration?Integrals are mathematical representations of numbers and functions that express notions such as volume, area, displacement, and other outcomes of the combination of little data. Finding integrals is the term used to describe the procedure.
By green's theorem -
[tex]\int\limits^{}_{} {} \, \int\limits^{}_{a} {5dA} \, = 5/3[/tex]
First, the integral given in this exercise corresponds to:
[tex]\int\limits^{}_{C} {((3y + 7e\sqrt{x}dx) +( 8x+ 5cos(y^2)) } \, dy[/tex]
Greens Theorem given as,
[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {(-\beta /\beta _{y} )P(x,y) + (\beta /\beta _{y} )Q(x,y)} \, dA[/tex]
and we have -
P(x,y) = 3y + [tex]7e^{\sqrt{x}}[/tex]
Q(x,y) = 8x + 5cos([tex]y^2[/tex])
And,
[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {5dA} \, dA[/tex]
From the question, our region is defined by:
lower bound: y= [tex]x^2[/tex]
upper bound: y = [tex]\sqrt{x}[/tex]
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Solve the equation -128 = 4x for x
Answer: -32
Step-by-step explanation:
To solve the equation -128 = 4x for x, we need to isolate the x variable on one side of the equation. We can do this by dividing both sides of the equation by 4. This gives us:
-128/4 = 4x/4
Dividing -128 by 4 gives us -32, so we have:
-32 = 4x/4
We can simplify this by dividing both sides by 4/4 to get:
-32 = x
Thus, the solution to the equation -128 = 4x is x = -32.
Two athletes start from the same point and move on a closed track of 600m. If they run in the same
direction at speeds 20m/s and 30m/s, when will they cross each other?
Mark one or more correct options
(A) 45 sec
(B) 1 min
(C) 1 min 30 sec
(D) 3 min
(E) 5 min
The solution is Option B , Option D , Option E
The athletes with speeds 20m/s and 30m/s will meet at every minute
What is Speed?
Speed is defined as the rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude
Speed = Distance / Time
Given data ,
Let the equation be represented as A
Now , the value of A is
The distance of the track is D = 600 m
Let the first athlete be A
Let the second athlete be B
The speed of A = 20 m/s
The speed of B = 30 m/s
The time taken by A to complete 600 m = 600 / 20 = 30 seconds
The time taken by B to complete 600 m = 600 / 30 = 20 seconds
Now , the time at which A and B will cross each other is the LCM of their respective time
So , The LCM of 20 and 30 is = 60 seconds
Therefore , the athletes will meet together at every 60 seconds or every 1 minute
Hence , the athletes will meet at every minute
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What is the y-intercept
write the prime factorization the foll numbers
A.144
Answer: 2 times 2 times 2 times 2 times 3 times 3
Step-by-step explanation:
Sam invests 1700 in a savings account that pays a nominal annual interest rate of 2.74% Sam makes no further payments or withdrawals from this account
David also invests 1700$ in a savings account that pays an annual rate of r% compounded yearly David makes no further payments or withdrawals from this account fund the value of r so that the amount in David’s account after 10 years will be equal to sams account
a.) The amount that Sammy will have in his account after 10 years will be= $2,165.8
$2,165.8b.) The value of 'r' required would be = 0.27%
What is interest rate?Interest rate is defined as the rate at which an individual receives an amount of money from an investment made over a period of time.
The amount invested by Sam(P) = $1700
The interest rate(R) = 2.74%
The time of investment (T) = 10 years
Simple interest = P×T×R/100
= 1700×10×2.74/100
= 46580/100
= $465.80
Therefore, the amount that Sammy will have in his account after 10 years will be = $1700 + $465.80
= $2,165.8.
For David's account interest rate the following is carried out:
The amount invested by Sam(P) = $1700
The interest rate(R) = r%
The time of investment (T) = 10 years
Simple interest (SI) = $465.80
Using the formula for simple interest;
SI = P×T×R/100
Make R the subject of formula;
R = SI×100/P×T
R = 465.80×100/1700×100
R= 46580/170000
R= 0.27%
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Complete question:
Sam invests $1700 in a saving account that pays a nominal annual rate of interest of 2.74% compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account. (a) Find the amount that Sam will have in his account after 10 years.
David also invests $1700 in a savings account that plays an annual rate of interest of r%, compounded yearly. David makes no further payments or withdrawals from this account. (b) Find the value of 'r' required so that the amount in David's account after 10 years will be equal to the amount in Sam's account.
two tangents $\overline{pa}$ and $\overline{pb}$ are drawn to a circle, where $p$ lies outside the circle, and $a$ and $b$ lie on the circle. the length of $\overline{pa}$ is $12,$ and the circle has a radius of $9.$ find the length $ab.$
The value of tangent AB as calculated from the given data is 14.4.
The tangents to the circle as given are PA and PB.
Let the point of intersection of PO and AB be X.
PA = 12
OA = 9
As given that the triangle is right angled we can use hypotenuse theorem to calculate PO,
PO = √ PA² + OA²
= √ 144 + 81
= 15
Now , the given triangles are similar because,
AX / AO = PA / PO
AX / 9 = 12 / 15
Therefore, the value of X is,
X = 7.2
Now we know that AB's midpoint is X,
Hence , AB = 2X = 14.4
Tangent is known as a point which passes through a circle only at one single point A circle can have infinite tangents.
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Suppose previous research suggests that the mean length of all adult Anacondas is 13 feet with a standard deviation of 2.4 feet. Let W be the random variable that stands for length of adult Anacondas, so E(W)=13, SD(W)=2.4. You are planning on collecting a random sample of 50 adult Anacondas. Consider the RV Bar-W, which is the mean of the 50 sampled Anacondas. 98% of samples will have the realized value of Bar-W less than what value? Which of the answers reasonably approximates the requested value of the sample mean with justification?a. "Bar-w"=14.67 since we can use a normal approximation by the CLTb. "Bar-w"=17.93 since we can use a normal approximation by the CLTc. "Bar-w"=13.70 since we can use a normal approximation by the CLTd. "Bar-w"=12.30 since we can use a normal approximation by the CLTe.A normal approximation is inappropriate
The answers reasonably approximates the requested value of the sample mean with justification is c. "Bar-w"=13.70 since we can use a normal approximation by the CLT.
Since the sample size is large (n=50), we can use the central limit theorem (CLT) to approximate the sampling distribution of the sample mean with a normal distribution.
The mean of the sampling distribution of the sample mean is equal to the mean of the population, which is 13 feet in this case. The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size. This is known as the standard error of the mean, and is denoted by SE(Bar-W).
In this case, SE(Bar-W) = 2.4/sqrt(50) = 0.48.
To find the 98th percentile of a normal distribution with mean 13 and standard deviation 0.48, we can use a standard normal table or a calculator to find that the 98th percentile is approximately 2.05.
Therefore, the value that 98% of samples will have the realized value of Bar-W less than is approximately 13 + 2.05 * 0.48 = 13.70. This means that the correct answer is (c) "Bar-w"=13.70 since we can use a normal approximation by the CLT.
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If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?
The distribution of Z = X/Y is λ₁ / (λ₁+λ₂).
What is a cumulative distribution function?The probability distribution of random variables is described using the cumulative distribution function. The probability for a discrete, continuous, or mixed variable may be described using it. The cumulative probability for a random variable is calculated by adding the probability density function.
Given:
X and Y are independent exponential random variables with respective parameters λ₁ and λ₂.
To find the distribution of Z = X/Y:
First, we have
[tex]f_x_y[/tex](x,y) = λ₁λ₂[tex]e^{-\lambda_1x[/tex][tex]e^{-\lambda_2y[/tex]
First, we find the cumulative distribution function (CDF) for Z = X/Y.
Derivative of Z, f(z) and put a = x/y
[tex]F_z[/tex](a) = P (X/Y ≤ a)
= P (X ≤ aY)
= [tex]\int\limits^{\infty}_ {x} \ \int\limits^{ay}_0 {\lambda_1\lambda_2e^{-\lambda_1x}e^{-\lambda_2y}\ dx dy[/tex]
[tex]= \int\limits^{\infty}_ {0} {\lambda_1\lambda_2e^{-\lambda_2y}\ dy [ -1/{\lambda_1}{e^{-\lambda_1x}]\limits^{ay}_ {0}[/tex]
[tex]= \lambda_2\int\limits^{\infty}_ {0} {e^{-\lambda_2y} - e^{-y(\lambda_2 + \lambda_1a)} \ dy[/tex]
[tex]= \lambda_2[ [{1/{\lambda_2}+{\lambda_1a]-{e^{-\lambda_2y} + e^{-y(\lambda_2 + \lambda_1a)}]\limits^{\infty}_ {0}[/tex]
[tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1) - 1][/tex]
[tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1a) - 1][/tex]
[tex]= [({\lambda_1a}/{\lambda_2 + \lambda_1a)][/tex]
So, P(X<Y) = P (X/Y <1)
= λ₁ / (λ₁+λ₂)
Therefore, distribution is λ₁ / (λ₁+λ₂).
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use the function f and the given real number a to find (f -1)'(a). (hint: see example 5. if an answer does not exist, enter dne.) f(x)
The function y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5 the real number is [tex]\frac{1}{6}[/tex].
y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5.
The slope of inverse functions are reciprocals at their corresponding points that is,
[tex]f^{-1}'(a)=\frac{1}{f'(b)}[/tex]
Where, [tex]f^{-1}'(a) = b[/tex] and f(b) = a
Now, determine the value of b for a= -5 using f(x)=[tex]x^{3}+3x-1[/tex]
f(b) = a
[tex]b^{3}+3b-1=-5[/tex]
[tex]b^{3}+3b= -4[/tex]
[tex]b = -1[/tex]
Therefore, [tex]f^{-1}'(a) = \frac{1}{f'(b)}[/tex]
[tex]f^{-1}'(-5)=\frac{1}{f'(-1)}[/tex]
Now, find f'(x) and evaluate it at
x= -1
f(x)=[tex]x^{3}+3x-1[/tex]
[tex]f'(x)= 3x^{2}+3[/tex]
f'(x)= [tex]3(x^{2} +1)[/tex]
then, f'(-1)= 3(1+1) = 6
Therefore, [tex]f^{-1}(-5)=\frac{1}{f^{-1}(-1)}=\frac{1}{6}[/tex].
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Suppose two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.
(a) Find the probability mass function for X.
(b) Find the expected value E(X).
(c) Find the variance V(X).
The expected value E(X) is 7 and the value of variance of X is 5.8333.
Two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.
a) The probability of mass function is obtained below:
The possible outcomes in each of the dice are 1 to 6. Therefore, the possible outcomes when two dice is rolled is 36
The sample space, s for fair dice (red die and blue die) is given below:
N(s) = 36
From the given information, two dice are rolled let X be the random variable measuring the sum of the two numbers rolled.
b) The expected value is calculated below:
The probability mass function of X is,
The required mean is,
E(x) = ∑xP(X=x)
[tex]=[2[/tex]×[tex]\frac{1}{36}+3[/tex]×[tex]\frac{2}{36}[/tex]+....+11×[tex]\frac{2}{36}+12[/tex]×[tex](\frac{1}{36} )[/tex]]
=[0.0556+0.1667+0.3333+0.5556+0.8333+1.1111+1.000+0.8333+0.6111+0.3333
=7
C) The variance V(X) is calculated below:
The probability mass function of X is,
The required variance is,
V(x)= ∑[tex](X-x)^{2}P(x) = 5.8333[/tex]
Therefore, the expected value E(X) is 7 and the value of variance of X is 5.8333.
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X-3y=12 what does it look like on the graph
The graph of x-3y = 12 is given.
What is graph?A graph is a visual representation or diagram that displays facts or values in an organized manner in mathematics. The points on a graph are typically used to depict the relationships between two or more things.
The equation is
x-3y = 12.
The graph of this line intersects the x-axis and y-axis.
And end behavior is infinity on both positive and negative.
And the graph is a straight line graph.
Therefore, behavior of x -3y = 12 is given in the graph.
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determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The fifth percentile of the standard normal distribution is -1.64.
In the given question, we have to determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The standard normal distribution, commonly known as the z-distribution, is a unique type of normal distribution in which the mean and standard deviation are both equal to 1. Any normal distribution's values can be transformed into z scores to normalise it. Z score shows the amount of standard deviations from the mean that each consists of numbers.
Since, X is normally distributed.
P(X< c) = 0.05
Using Z table
c = -1.645
Fifth percentile = -1.64
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Which expression is equivalent to (8×10−2) (2.4×10−3)?
The expression that is equivalent to the given expression, (8×10−2) (2.4×10−3), is 1.92 × 10⁻⁴
Determining an equivalent expressionFrom the question, we are to determine the expression that is equivalent to the given expression
From the given information,
The given expression is
(8×10−2) (2.4×10−3)
First, we will write this expression properly
The given expression written properly is
(8 × 10⁻²)(2.4 × 10⁻³)
Now, we will evaluate the expression
Evaluating the expression
(8 × 10⁻²)(2.4 × 10⁻³)
8 × 2.4 × 10⁻² × 10⁻³
19.2 × 10⁻⁵
= 1.92 × 10⁻⁴
Hence, the expression is 1.92 × 10⁻⁴
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f the temperature is 50 degrees Fahrenheit, what is the temperature in Celsius?
Answer:
the answer is 10 degrees Celsius
50 Degrees Fahrenheit is 10 degrees Celsius.
Here is the formula
(50°F − 32) × 5/9 = 10°C
What rigid motion maps the solid-line figure onto the dotted-line figure?
A reflection
B. rotation
C. translation
Answer:
A. Reflection
Step-by-step explanation:
I believe the answer would be reflection
Realiza las siguientes potencias
Answer:
-5/8
Step-by-step explanation:
5.(-2^-3)
-5. 1/2³
-5. 1/8
-5/8
The best approximation, in cubic inches, to the increase in volume of a sphere when the radius is increased from 3 to 3.1 inches, is Use V = Tr3 (a) 0.04n/3 (b) 3.61 n'
(c) 0.04 n'
(d) 1.2 n'
(e)36 n'
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
How is volume of a sphere calculated?In simple words, when a sphere is placed in the space or three-dimension space, then the total space acquired by the sphere generally shows the volume of the sphere. The mathematical formula needed to determine the volume of a sphere is shown below,
[tex]V = \frac{4}{3} \pi r^3[/tex]
Calculation:Given- The initial radius of the sphere is r1 = 3 inch and the final radius of the sphere is r2 = 3.1 inch.
Now , the change in the radius of the sphere is [tex]dR =[/tex][tex]r2 - r1 =\\[/tex] 3.1 - 3 = 0.1 inch.
The mathematical expression of the volume of a sphere is shown below,
[tex]V = \frac{4}{3}\pi r^3[/tex] (here, [tex]r[/tex] is the radius of sphere)
Differentiate the above shown expression with respect to radius ([tex]r[/tex]) as shown below,
[tex]\frac{d}{dr} (V) = \frac{d}{dr} (\frac{4}{3} ) \pi r^3[/tex]
∴
dV/dR = 4/3 π × d/dR(r^3)
= 4/3 π × 3r^2
dV/dR = 4π r^2
∴ dV = 4π r^2 dR ................(1)
Substitute all values in equation (1) and calculate the approximation as shown below,
dV = 4π r^2 dR
= 4π (3)^2 × (0.1)
∴ dV = 3.6π cubic inches.
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
an =
n sqr root 35 + 3n
lim nâ[infinity] an =
This sequence is not convergent because the nth term does not approach a single limit as n approaches infinity. The limit does not exist, so the answer is DNE.
This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. This means that the value of an does not approach a single limit as n approaches infinity, and so the limit does not exist. Therefore, the answer for the limit as n approaches infinity is DNE.
This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. As n increases, the value of an increases, and so the value of an does not approach a single limit as n approaches infinity. This means that the limit does not exist, and so the answer for the limit as n approaches infinity is DNE.
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A rectangular park is 4a m and 3a m broad, find it's area in m2
Answer:
Area = 12a m²
Step-by-step explanation:
Given information,
→ Length = 4a m
→ Width = 3a m
Now we have to,
→ find the area of rectangular park.
Formula we use,
→ Area = L × W
Then the area of rectangle is,
→ L × W
→ 4a × 3a
→ (4 × 3)a
→ 12a m²
Therefore, the area is 12a m².
An airline that wants to assess customer satisfaction chooses a random sample of 10 of its flights during a single month and asks all of the passengers on those flights to fill out a survey. This is an example of a
A) Multistage sample.
B) Stratified sample.
C) Cluster sample.
D) Simple random sample.
E) Convenience sample
An airline that wishes to measure customer satisfaction chooses ten of its flights at random over the course of a single month and requests that each passenger fill out a survey. A Cluster sample looks like this.
Why do we utilize cluster sampling?Cluster sampling should be used to study large, dispersed populations because interviewing every person would be costly, time-consuming, and occasionally impossible.Smaller, more comparable groups that represent the population under study can be produced via cluster sampling.You divide a population into groups, such as districts or schools, and then randomly select a sample from among these groupings when using the probability sampling approach known as cluster sampling. Each cluster should ideally be a miniature representation of the total population.An airline that wants to gauge customer happiness selects a haphazard sample of 10 of its flights over the course of a single month and asks each passenger to complete a survey. An illustration of a Cluster sample is this.To learn more about Cluster sample refer to:
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small regional carrier accepted 22 reservations for a particular flight with 18 seats. 16 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 58% chance, independently of each other.
(Report answers accurate to 4 decimal places.)
Find the probability that overbooking occurs. Find the probability that the flight has empty seats.
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
How to calculate probability?We remove the 16 regular customers and their seats from consideration. Thus, we consider the 22 - 16 remaining customers = 6 and the 18-16 = 2 seats overbooking means 3 or more of the 6 remaining passengers; each has a 58% chance remaining. We can either use the binomial distribution formula P(x) = C(6, x) .58^x .42^(6-x) or use packages that provide the result, including cumulative probabilities.
Calculation:Overbooking = 1 - P(X <= 2) In Excel, we use =1 - binom.dist(2,6,.58,1) ; the 1 is for cumulative.
The solution is 0.7920
Probability of empty seats = P(x <= 1) = binom.dist(1,6,.58,1) = 0.0510
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
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Hummingbird feeders are filled with hummingbird nectar. Hummingbird
nectar is made using a ratio of 1 part sugar to 3 parts water.
How much sugar, in ounces, is needed to make 28 ounces of
hummingbird nectar? Show your work in the provided space.
The amount of sugar, in ounces, that is needed to make 28 ounces of hummingbird nectar is 7 ounces.
How to find the ounces of sugar needed ?Hummingbird nectar is said to be made by using a ratio of 1 part sugar to 3 parts water. For every 1 ounce of sugar therefore, there would be 3 ounces of water.
Put together, this would give 4 ounces. And the percentage of sugar in this 4 ounces would be:
= 1 / 4
= 25 %
If 28 ounces of hummingbird nectar needed to be made, the sugar needed is :
= Percentage of sugar x Ounces of hummingbird nectar
= 25 % x 28
= 7 ounces of sugar
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Q. An arithmetic series has first term a and common difference d, where d is a prime number. The sum of the first n terms of the series is S, and Sm=39 S2m = 320 Find the value of d and the value of m Show clear algebraic working. (Total for question = 5 marks)
Finding the values of d and the value of m with the algebraic working will give us the the value of d to be 7 and the value of m to be 3.
How do we calculate the values using the algebraic expression?Finding the value of d, we know that the sum of the first n terms of an arithmetic series is given by:
S = n/2 * (2a + (n-1)d)
Since the sum of the first n terms is S and the sum of the first 2m terms is S2m, we can set up the following equation:
S = m/2 * (2a + (m-1)d)
S2m = 2m/2 * (2a + (2m-1)d)
Substituting the given values for S and S2m into these equations, we get:
39 = m/2 * (2a + (m-1)d)
320 = 2m/2 * (2a + (2m-1)d)
Solving for d in each equation, we find that d = -7 in the first equation and d = 7 in the second equation. Since d must be a prime number, the only possible value for d is 7.
Now that we know the value of d, we can solve for m. Substituting the value of d back into one of the equations and solving for m, we get:
39 = m/2 * (2a + (m-1)7)
78 = m * (2a + (m-1)7)
78 = m * 2a + 7m^2 - 7m
7m^2 - m - 78 = 0
We can solve for m using the quadratic formula:
m = (-1 +/- sqrt(1^2 - 4*7*(-78)))/(2*7)
= (-1 +/- sqrt(2521))/14
= (-1 + 49)/14 = 3
= (-1 - 49)/14 = -7
Since m must be a positive integer, the only possible value for m is 3.
Therefore, the value of d is 7 and the value of m is 3.
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The diagonal of a rectangular room is 13 ft long. One wall measures 7ft longer than the adjacent wall. Find the dimensions of the room.
Let x be the length of the shorter wall of the rectangular room and y be the length of the longer wall of the rectangular room. Since the longer wall is 7 ft longer than the shorter wall, we can write the equation y = x + 7. We can also express the diagonal of the rectangular room using the Pythagorean theorem as x^2 + y^2 = 13^2. We can solve for x and y by substituting the equation y = x + 7 into the equation x^2 + y^2 = 13^2 and then solving for x. Doing this, we get x^2 + (x + 7)^2 = 13^2. Expanding the square on the right side of the equation and then rearranging the terms, we get x^2 + 2x^2 + 14x + 49 = 169. Combining like terms, we get 3x^2 + 14x - 120 = 0. This quadratic equation can be factored as (x - 8)(3x + 15) = 0. Since the length of a side of a rectangle must be positive, we can ignore the solution x = -15/3. So, the length of the shorter wall of the rectangular room is x = 8 ft. The length of the longer wall can be found by substituting this value into the equation y = x + 7, giving us y = 8 + 7 = 15 ft. Therefore, the dimensions of the rectangular room are 8 ft by 15 ft.
The measures of the angles of a triangle are shown in the figure below. Solve for x.
to
76⁰
41
The measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.
What is Angle Sum Property?
The sum of all angles of a triangle is equal to the angle of a straight line i.e. 180°. If we have a triangle ABC, then the Sum of angles A , B, and angle C will be 180 ° and the value of the exterior angle is equal to the sum of two interior opposite angle.
We have,
The measures of the angles of a triangle are:
x,
76⁰,
41⁰.
so, we have to calculate the value of the x⁰:
The sum of all the angles in the triangle is 180⁰
so,
to calculate the measure of x put the all values in equation such that using the above rule:
x + 76⁰ + 41⁰ = 180⁰
x = 180⁰ - 76⁰ - 41⁰
x = 63⁰
Hence, the measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.
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A hot air balloon travels 2016 miles in 96 hours. The balloon travels the same number of miles each hour. How many miles does the balloon travel in 1 hour
Answer:
21 m/ph
Step-by-step explanation:
2016 divided by 96
The balloon travels 21 miles in one hour.
What is division?In mathematics, division is one kind of operation. The phrases or numbers in this process are split into the same number of parts.
Given, a hot air balloon travels 2016 miles in 96 hours.
And the balloon travels the same number of miles each hour.
To find the same numbers of miles per hour:
We use the division of 2016 by 96.
That means,
2016 / 96
= 21
Therefore, the travel time of the balloon is 21 miles per hour.
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Could I get some help with this? Thanks! Here’s the question to go along with it.
For each diagram, calculate the value of x. Show your work and include an explanation of what you used (definitions and theorems) to solve the problem. If not possible, state why.
Answer:
x = 16 1/3°
Step-by-step explanation:
You have parallel lines with consecutive interior angles marked 7x-28° and 5x+12°, and you want to find the value of x.
Consecutive interior anglesAt parallel lines, consecutive interior angles, or same-side interior angles, are supplementary. This means the total of the two marked angles is 180°.
7x -28° +5x +12° = 180°
12x -16° = 180° . . . . . . . . . . simplify
12x = 196° . . . . . . . . . add 16°
x = (49/3)° = 16 1/3° . . . . . . . divide by 12
__
Additional comment
∠3 = 86 1/3°
∠5 = 93 2/3°
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.72.
(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 23 specimens from the seam was 4.85. (Round your answers to two decimal places.)
(b) Compute a 98% CI for true average porosity of another seam based on 14 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.)
(c) How large a sample size is necessary if the width of the 95% interval is to be 0.42? (Round your answer up to the nearest whole number.)
(d) What sample size is necessary to estimate true average porosity to within 0.24 with 99% confidence? (Round your answer up to the nearest whole number.)
For the normal distribution of helium porosity (in percentage) of coal samples,
a) 95% CI for the true average porosity of a certain seam is equals to ( 4.56 , 5.14 ).
b) A 98% CI for true average porosity of another seam based on 14 specimens is equals to (4.11 , 5.01).
c) Sample size with width of confidence interval 0.42 is 45.
d) Necessary Sample size to estimate true average porosity to within 0.24 is 60.
We have Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed.
Standard deviations, σ = 0.72
a)Average porosity of specimens , X-bar = 4.85
Sample size, n = 23
Confidence level , alpha = 95% = 0.95
From Normal Distribution table
Z for 95% Confidence Interval = 1.96
so,95% Confidence interval = (x-bar - Z× σ/sqrt(n) , x-bar + Z×σ/sqrt(n) )
plugging all known values in above formula,
= ( 4.85-1.96×0.72/sqrt(23) , 4.85+1.96×0.72/sqrt(23) )
So, 95% Confidence interval = ( 4.56 , 5.14 )
b)Now, Sample size, n = 14
sample mean, X-bar = 4.56
from normal distribution Z- table
Z-score for 98% CI is equal to 2.33
so,98% CI = (x-bar-Z×σ/sqrt(n) , x-bar+Z×σ/sqrt(n) )
= ( 4.56-2.33×0.72/√14 , 4.56+2.33×0.72/√14 )
98% CI = ( 4.11 , 5.01 )
c)Using the normal Distribution Z- table
Z for 95% CI = 1.96
Width of confidence interval = 0.42
we have to determine the sample size, n .
Using formula, Width = 2×Z×σ/sqrt(n)
=> n = (2×Z×sd/width)²
=> n= (2×1.96×0.72/0.42)²
=> n = 45.15
=> n = 45 ( whole number)
d) From normal distribution table
Z for 99% CI = 2.58
Margin of error, ME = 0.24
margin of error = ME = Z×σ/sqrt(n)
=> n = (Z× σ/ME)²
=> n= (2.58×0.72/0.24)²
=> n = 59.9
=> n = 60
Hence, required sample size is 60.
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