Let x be the number of gallons Mrs M can purchase;
Given information:
Price per gallon: $3.50
Mrs M. Total budget: $10.50
[tex]\begin{gathered} 3.50x\text{ }≤\text{ }10.50 \\ \end{gathered}[/tex]Solving for the number of gallons Mrs M can purchase:
[tex]\begin{gathered} x\leq\frac{10.50}{3.50} \\ \\ x\text{ }\operatorname{\leq}\text{ }3 \end{gathered}[/tex]ANSWER
a. g>0 and g<3
The weights of a certain brand of candies are normally distributed with a mean weight of 0.8543 g and a standard deviation of 0.0519 g. A sample of these candies came from a package containing 469 candies, and the package label stated that the net weight is 400.3 g. (If every package has 469 candies, the mean weight of the candies must exceed
400.3
469=0.8536 g for the net contents to weigh at least 400.3 g.)
Given,
The weight of candies normally distributed ;
Mean weight of candies = 0.8543 g
Standard deviation, σ = 0.0519 g
Sample of candy came from a packet of 469 candies
Net weight of the packet = 400.3 g
Average weight of the candies in the packet;
[tex]x_{avg} =[/tex] ∑xi/n = 400.3/469 = 0.8535
The population standard deviation (sigma=0.0519 g) is the standard deviation for a confectionery that was chosen at random (sample size n=1).
The z-score can be used to determine the likelihood that an element has a weight greater than 0.8536z = (X - μ) / (σ/√n) = (0.8536 - 0.8535) / (0.0519/√1) = 0.0001/0.0519 = 0.002
P(X > 0.8536) = P(z > 0.002) = 0.4992
A randomly chosen candy has a probability P=0.4992 of weighing at least 0.8536 g.
The z-score needs to be computed if the sample now has n=441 candies and we want to know the likelihood that the mean weight is at least 0.8543 g:z = (X - μ) / (σ/√n) = (0.8543 - 0.8535) / (0.0519/√441) = 0.0008/0.0024 = 0.3288
P(X > 0.8543) = P(z > 0.3288) = 0.37115
A sample of 441 candies chosen at random has a probability P=0.3712 of having an average weight of at least 0.8543 g.
We may determine the likelihood that, for a package of 469 candies and using the mean of 0.8535 g that we previously determined, the average weight is at least 0.8556 g in order to be more certain of the claim that the mean weight is 0.8556 g.z = (X - μ) / (σ/√n) = (0.8556 - 0.8535) / (0.0519/√469) = 0.0021/0.0024 = 0.89
P(X > 0.8556) = P(z > 0.89) = 0.18673
Given that the likelihood is P=0.187, the brand's claim that it delivers the amount promised to customers on the label needs to be reevaluated.
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The population of a certain town was 10,000 in 1990. The rate of change of the population, measured in people per year, is modeled by P prime of t equals two-hundred times e to the 0.02t power, where t is measured in years since 1990. Discuss the meaning of the integral from zero to twenty of P prime of t, d t. Calculate the change in population between 1995 and 2000. Do we have enough information to calculate the population in 2020? If so, what is the population in 2020? Explain your answers.
Given the function of the rate of change of the population:
[tex]P(t)=200e^{0.02t}[/tex]The integral of this function will be the value of the population growth of the town from 1990 to 2010. This is because P(t) is the rate of change of population, its integral will be the antiderivative, that is the population of the town.
To calculate the change in population we have to find the integral from 5 to 10 of P(t):
[tex]\begin{gathered} \int_5^{10}200e^{0.02t}dt \\ 200\int_5^{10}e^{0.02t}dt \\ 200\cdot\frac{e^{0.02t}}{0.02} \\ 10000e^{0.02t} \\ (10000e^{0.02(10)})-(10000e^{0.02(5)}) \\ 1162.3 \end{gathered}[/tex]The change in population can not be a decimal number, so we can round it to 1162.
To calculate the population in 2020 we have to find the integral from 0 to 30 of P(t) and then add it to 10000 which was the initial population (we already know the integral so we're just going to evaluate it):
[tex]\begin{gathered} 10000e^{0.02(30)}-10000e^{0.02(0)} \\ 8221.2 \\ 10000+8221.2=18221.2 \end{gathered}[/tex]It means that the population in 2020 is 18221 (Remember that we round it because it is not possible to have a decimal value for the population of the town).
Question 6Three times the sum of a number and four is equal to five timesthe number, decreased by two. If x represents the number, whichis a correct translation of the statement?A3(x + 4) = 5x - 23x + 4 = 5x - 2B3(x + 4) = 5(x - 2)D3x + 4 = 5(x - 2)
'the sum of a number and four' means
[tex](x+4)[/tex]And three times that is
[tex]3(x+4)[/tex]Then, that 'equals to five times a number'
[tex]5x[/tex]'decreased by two':
[tex]5x-2[/tex]So, the statement is translated into the equation:
[tex]3(x+4)=5x-2[/tex]The correct answer is option A
Is (-2,-6) a solution to the equation y = 3x?
Oyes
Ono
Answer:
yes
Step-by-step explanation:
if y = -6,
3x = -6
x = -2
Answer:
yes
Step-by-step explanation:
y = 3x?
Let x = -2 and y = -6
-6 = 3(-2)
-6 = -6
This is a true statement so (-2,-6) is a solution to the equation y = 3x
Find the critical value (tα/2) for a 99% confidence interval if the sample size is 15. Round your answer to three decimal places.
tα/2 =
The critical value (tα/2) for a 99% confidence interval for the sample size 15 is 2.977 .
We have to find the critical value tα/2 for 99% confidence interval, we are given the sample size.
First we will find the alpha value to get the critical value.
=100%-99%
=1-0.99
=0.01
α=0.01
α/2=0.01/2
α/2=0.005
degrees of freedom(df)= n-1=15-1=14
We will use degrees of freedom and α/2 values to find the critical value that is tα/2.
By using the t table we get the critical value of tα/2=2.977.
Therefore, the critical value for 99% confidence interval with sample size 15 is 2.977.
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4x^3 + 12x^2 + 3x + 9 factor completely
The factored expression of 4x^3 + 12x^2 + 3x + 9 is (4x^2 + 3)(x + 3)
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to factor the expression completely?The expression is given as
4x^3 + 12x^2 + 3x + 9
Group the expression in 2's
So, we have
4x^3 + 12x^2 + 3x + 9 = (4x^3 + 12x^2) + (3x + 9)
Factor each group
So, we have
4x^3 + 12x^2 + 3x + 9 = 4x^2(x + 3) + 3(x + 3)
Rewrite as
4x^3 + 12x^2 + 3x + 9 = (4x^2 + 3)(x + 3)
Hence, the factored expression is (4x^2 + 3)(x + 3)
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If f(x)=√x and g(x)= - 2x+3, find (f . g)(x) and (g . f)(x).
(f . g)(x)=_____
(Simplify your answer. Type an exact answer, using radicals as needed.)
Thee values of the functions are;
(f . g)(x) = √(-2x + 3)
(g . f)(x) = - 2√x + 3
What is a function?A function can simply be defined as a law, rule or expression showing the relationship between two variables in an equation or expression.
The two variables are given;
The dependent variableThe independent variableGiven the functions;
f(x)=√x g(x)= - 2x+3To determine the function (f . g)(x), substitute the value of x as g(x) in the function, we have;
(f . g)(x) = √(-2x + 3)
To determine the function (g . f)(x), substitute the values of x in the function g(x), we have;
(g . f)(x) = - 2√x + 3
Hence, the functions are √(-2x + 3) and - 2√x + 3
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I need help with this question parts a - e
Given:
650 lb of coffee is sold when the price is $4 per pound,
400 lb of coffee is sold at $8 per pound.
To find:
A) List the data points
B) The slope of line joining the points
C) Interpretation of meaning of slope of line
D) linear equation of data using point slope form
E) Using the function we need to predict the number of consumers buying at $6 per pound
Step-by-step solution:
A) Data points of the given data:
Data points:
(4,650) and (8,400)
B) The slope of the line joining the points:
[tex]\begin{gathered} Slope=\frac{y_2-y_1}{x_2-x_1} \\ \\ Slope=\frac{400-650}{8-4} \\ \\ Slope\text{ =-62.5} \end{gathered}[/tex]C) Interpretation of meaning of slope of line:
Slope here means that:
When there is an increase in the price of coffee then the weight of coffee decreases.
Weight per dollar of coffee decreases at the rate of 62.5
D) linear equation of data using the point-slope form:
Applying point-slope form:
y - y1 = m(x - x1)
y - 650 = -62.5(x - 4)
y = -62.5x + 900
In function notation it can be represented as:
S(P) = -62.5P + 900
E) Using the function we need to predict the number of consumers buying at $6 per pound
To predict that, we will simply put the value of P equal to 6 in the given function:
S(P) = -62.5P + 900 At P = 6
S(P) = -62.5(6) + 900
S(P) = 525
Thus we can say that 525 customers will buy at $6 per pound.
Find the solution to the system of equations represented by the matrix shownbelow.83-161E441371-63ХUse the operation buttons to start solving
The solution to the system of equations represented by the matrix is (x, y, z) = (6, 7, 6).
Maybe your matrix is ...
[tex]$\left[\begin{array}{ccc|c}3 & -1 & 7 & 53 \\ 1 & 7 & 1 & 61 \\ 9 & 1 & 1 & 67\end{array}\right]$[/tex]
A calculator can tell you the solution is ...
(x, y, z) = (6, 7, 6)
The third variable in systems of equations with more than two variables can be defined in terms of the other two (as for solution by substitution). This can be used to create two equations with two unknowns by substituting it into the remaining equations. The value of the third variable can then be determined using that answer. The use of this technology is demonstrated in the attached.
The last equation served as a definition, which was then applied to the first two equations. An algebraic solution can be solved using the same strategy.
The answer is (x, y, and z) = (6, 7, 6).
Complete question: Find the solution to the system of equations represented by the matrix shown below. 3 -1 7 53 1 7 1 61 9 1 1 67
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Steve files head of household. In 2022, he received $23,000 in social security benefits, $10,000 in taxable retirement income, and $4,000 in interest and dividend income. Using the Social Security Benefits Worksheet - Line 6a and 6b, determine Steve's taxable social security benefits.
Steve's taxable social security benefits is zero.
Since Steve declares himself as the head of the home, his social security payments, which range from $25,000 to $34,000, will be subject to a 50% tax.
Retirement income, interest income, and dividend income are not considered social security benefits for determining the limit. Nevertheless, in the case at hand, the social security benefits total $23,000, which is less than the maximum of $25,000 and is thus not taxable.
As a result, Steve's Social Security benefits are not taxable.
Hence the taxable amount on Steve on social security benefits is Zero.
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Put 3/4, -1/2, -1/4 in order from least to greatest.
Answer:
-1/2, -1/4, 3/4.
Explanation:
If we draw the number in a number line, we get:
So, if we order them from least to greatest, we get:
-1/2, -1/4, 3/4.
Answer:
-1/2, -1/4, 3/4
Step-by-step explanation:
Debbie's checkbook balance was $2,715.50. She wrote checks totaling $5,298.13 during the month of May. Deposits made during the month were $250.16 on May 1. On May 15, her payroll check was electronically processed for $2,980.66; on May 22, a check from the IRS was electronically processed for $2,550. On May 27, she recorded $534.10 for payments debited to her account for auto insurance and utilities. What is her checkbook balance?
Debbie's checkbook balance is $-3,297.23, which shows an overdraft.
When does an overdraft occur?An overdraft occurs when the money (deposits) in Debbie's account does not exceed her payments or withdrawals.
While Debbie had a positive beginning balance, the ending balance shows an overdraft of $-3,297.23.
By this overdraft, Debbie's bank is extending her some short-term credit, which she should pay back with interest.
Beginning balance on checkbook = $2,715.50
Checks for the month = $-5,298.13
Deposits for the month = $250.16
Payroll check = $-2,980.66
IRS check = $2,550
Auto insurance and utilities = $-534.10
Ending balance on checkbook = $-3,297.23
Thus, Debbie's ending checkbook balance will show an overdraft of $-3,297.23.
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8.26×10^0 scientific notation or standarn form
The number 8.26 10^0 is shown in scientific notation.
Its standrad form is: 8.26 since we know that any base different from zero raised to the power "0" (zero) gives "1" (one).
can you solve 3 5/12 + 1 3/8 ?
Answer:
4 19/24
Explanation:
First, we need to convert the mixed numbers into a fraction, so:
[tex]\begin{gathered} 3\frac{5}{12}=\frac{3\times12+5}{12}=\frac{36+5}{12}=\frac{41}{12} \\ 1\frac{3}{8}=\frac{1\times8+3}{8}=\frac{8+3}{8}=\frac{11}{8} \end{gathered}[/tex]Then, we can add the fractions as follows:
[tex]\frac{41}{12}+\frac{11}{8}=\frac{41(8)+12(11)}{12(8)}=\frac{328+132}{96}=\frac{460}{96}[/tex]Finally, we can simplify the fraction and write the fraction as a mixed number as:
[tex]\frac{460}{96}=\frac{460\div4}{96\div4}=\frac{115}{24}[/tex][tex]\frac{115}{24}=4\frac{19}{24}[/tex]Because when we divide 115 by 24, we get 4 as a quotient and 19 as a remainder.
Therefore, the answer is: 4 19/24
Write an inequality that represents the graph below
An inequality that represents the given graph is; 2 < x < ∞.
What is referred as the term inequality?In mathematics, an inequality is a link between two expressions as well as values that aren't equal to each other.Inequality results from a lack of balance. For instance, suppose you would like to buy a new vehicle that costs $250 but only have 225. It is also a inequality because you are comparing these two non-equal numbers.For the given question.
The inequality is given by the values of the number line.
The value is starting from 2 and goes up to infinity.
But, It is a hollow dot at 2 means 2 is with open interval, as 2 will not be taken for the value of x.
The values lies between, (2, ∞)
Thus, the inequality that represents the given graph is; 2 < x < ∞.
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17. The rent for a store is GH¢420.00 a year.
What is the rent for the store in six
months?
Answer:
GH¢210Step-by-step explanation:
The rent is 420 a year.
6 months is half a year.
The rent for 6 months is half the rent for a year:
420/2 = 210find inverse of function f(x)=3/x+2
What are the dimensions of the rectangle with Maximum area?
Let x be the length of the rectangle and y be the width of the rectangle, then we can set the following equations:
[tex]\begin{gathered} 2x+2y=40, \\ A=x\cdot y\text{.} \end{gathered}[/tex]Solving the first equation for x we get:
[tex]\begin{gathered} 2x=40-2y, \\ x=\frac{40}{2}-\frac{2y}{2}, \\ x=20-y\text{.} \end{gathered}[/tex]Substituting x=20-y in the second equation we get:
[tex]\begin{gathered} A=(20-y)\times y, \\ A=20y-y^2. \end{gathered}[/tex]Now, we will use the first and second derivative criteria to find the maximum.
The first and second derivatives are:
[tex]\begin{gathered} A^{\prime}(y)=20-2y. \\ A^{\prime\prime}(y)=-2. \end{gathered}[/tex]Since the second derivative is a negative number that means that A(y) reaches a maximum when A´(y)=0.
Solving A´(y)=0 for y we get:
[tex]\begin{gathered} 20-2y=0, \\ 20=2y, \\ 10=y\text{.} \end{gathered}[/tex]Now, substituting y=10 in x=20-y, we get:
[tex]x=20-10=10.[/tex]Answer:
Length 10 yards.
Width 10 yards.
Which equation represents a horizontal line that passes through the point (2, 7)? А x=2 В y = 2 C X= 7 D y=7
It is important to know that horizontal lines are represented by the equation y = k, where k is a real numbers.
In this case, if the equation passes thorugh (2,7), then the equation would be y = 7.
Hence, the answer is D.A train travels 150 km in 2 hours and 30 minutes. What is its average speed?
Answer:
It's average speed is 48 km per hour
Step-by-step explanation:
Answer:
60 km/hr
Step-by-step explanation:
150 km / 2.5 hr = 60 km/hr
What is 372.7332 rounded to the nearest thousandth?
Please help rn I have until 3pm
Rotate this hexagon 180º.
(0, -8)
(3, -8)
(6,-5)
(6, 1)
(3, 2)
(0, 2)
After rotation the coordinates of the hexagon is (0, 8), (-3, 8), (-6, 5), (-6, -1), (-3, -2), (0, -2)
Upon rotation of this hexagon by 180
The rule for a rotation by 180° about the origin is (x,y)→(−x,−y) .
After rotation, the co- ordinates change from
(0, -8) → (0, 8)
(3, -8) → (-3, 8)
(6,-5) → (-6, 5)
(6, 1) → (-6, -1)
(3, 2) → (-3, -2)
(0, 2) → (0, -2)
Therefore, after rotation the coordinates of the hexagon is (0, 8), (-3, 8), (-6, 5), (-6, -1), (-3, -2), (0, -2)
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The hypotenuse of right triangle XYZ is 28 and m
Solution
For this case we can do the following:
m < X = 60
One angle in a right triangle needs to be 90º
So then we can find the measure of the other angle like this.
180 -90-60 = 30º
We can find the following identity:
[tex]\sin 60=\frac{y}{28}[/tex]And solving for thw leg opposite we got:
[tex]y=28\cdot\sin 60=24.25[/tex]
I am having trouble with the last two questions. I worked through finding the slope and y-intercept with a previous tutor, before we got disconnected.6. Express this equation as a function D of P and find its domain.7. How many shirts per month will be demanded if the price is $27?
Data:
[tex]m=-\frac{203}{152}\approx-1.3355[/tex][tex]b=\frac{11227}{152}\approx73.862[/tex]Equation as a function of D of P
[tex]D(P)=-1.3355\cdot P+73.862[/tex]Answer (6):
• Domain
[tex](-\infty,+\infty)[/tex]To know how many shirts per month will be demanded if the price is $27, we have to use P = $27:
[tex]D(27)=-1.3355\cdot27+73.862\text{ }\approx37.80[/tex]Answer (7): $37.80
Graph the solution of the following inequality and answer the questions below
We have the next inequality
[tex]11x+9<-11y+9[/tex]First we need to isolate the y
[tex]11y<-11x+9-9[/tex][tex]11y<-11x[/tex][tex]y<\frac{-11}{11}x[/tex][tex]y<-x[/tex]The boundary line need to be dashed because the sign is less than
Two points of the boundary line can be calculated using the next equality y=-x
x=0
y=0
Point (0,0)
x=-5
y=5
Point (-5,5)
The area will be the area below the boundary line because we have a sign less than
In isosceles △ABC where AC≅BC, altitiude CD is drawn. If AC= 17 and AB= 30. Determine the altitiude of the triangle.
By definition, an Isosceles triangle is a triangle that have two congruent sides, and the altitude divides the triangle into two equal Right triangles.
Remember that a Right triangle is a triangle that has an angle whose measure is 90 degrees.
Based on this, you know that:
[tex]\begin{gathered} AD=BD=\frac{AB}{2} \\ \\ AC=BC \end{gathered}[/tex]Knowing that:
[tex]\begin{gathered} AB=30 \\ AC=17 \end{gathered}[/tex]You get that:
[tex]\begin{gathered} AD=BD=\frac{30}{2}=15 \\ \\ AC=BC=17 \end{gathered}[/tex]The Pythagorean Theorem states that:
[tex]a^2=b^2+c^2[/tex]Where "a" is the hypotenuse and "b" and "c" are the legs of the right triangle.
So you can identify that, for this case:
[tex]\begin{gathered} a=17 \\ b=15 \\ c=CD \end{gathered}[/tex]Where "CD" is the altitude of the triangle.
Therefore, substituting values and solving for "CD", you get this result:
[tex]\begin{gathered} 17^2=15^2+CD^2 \\ \sqrt[]{17^2-15^2}=CD \\ CD=8 \end{gathered}[/tex]The answer is: Option (1)
16. A rectangular picture measures 6 inches by 8 inches. Simon wants to build a
wooden frame for the picture so that the framed picture takes up a maximum area
of 100 square inches on his wall. The pieces of wood that he uses to build the
frame all have the same width. Write an equation or inequality that could be used
to determine the maximum width of the pieces of wood for the frame Simon could
create.
Explain how your equation or inequality models the situation.
Solve the equation or inequality to determine the maximum width of the pieces of
wood used for the frame to the nearest tenth of an inch.
Answer:
[tex]w \leq 1.9[/tex]
Step-by-step explanation:
The picture itself measures 6" by 8". Therefore, the area is 6 * 8 = 48 square inches.
All of the pieces of wood have the same width, and we know that two of them will be 6 inches long and two will be 8 inches long.
Therefore, the combined area of the pieces of wood will be 2 * 6w and 2 * 8w. This gives us 12w + 16w.
Now we can set up an inequality:
48 (area of picture) + 12w + 16w <= 100 (maximum area).
Simplify the left side of the equation to give us this:
[tex]48+28w\leq 100.[/tex]
We can move the 48 over to the other side to get [tex]28w \leq 100 - 48[/tex], which simplifies to [tex]28w \leq 56[/tex]. Now, you just have to divide both sides by 28 to get [tex]w\leq 1.857[/tex].
Rounding to the nearest tenth of an inch gives us 1.9 as the maximum width of the pieces of wood.
Nandita earned 224 dollars last month. she earned 28 dollars by selling cards at a craft fair and the rest of the money by babysitting. Complete an equation that models the situation and can be used to determine x, the number of dollars Nandita earned last month by babysitting.
the selling price of each card is = 28 $
let she sold x number of cards,
so, the equation is,
28x = 224
x = 224/28
x = 8
thus, the answer is number of cards sold by nandita is 8
so, the equation is
28x = 224
Determine the open intervals on which the function is increasing, decreasing, or constant.(Enter your answers using interval notation. If an answer does not exist, enter DNE.)
Given:
[tex]f(x)=\sqrt{x^2-4}[/tex]To find:
The interval at which the function is increasing, decreasing and constant.
Explanation:
We know that,
For a function, y = F(x), if the value of y is increasing on increasing the value of x, then the function is known as an increasing function.
For a function, y = F(x), if the value of y is decreasing on increasing the value of x, then the function is known as a decreasing function.
According to the graph,
The function is increasing in the interval,
[tex][2,\infty)[/tex]Because, if x increases from 2, the value of y increases.
The function is decreasing in the interval,
[tex](-\infty,-2][/tex]Because, if x increases from negative infinity, the value of y decreases.
As we know, a constant function is a function whose output value is the same for every input value.
Here, the function is not constant at any of the intervals.
Final answer:
Increasing:
[tex][2,\infty)[/tex]Decreasing:
[tex](-\infty,-2][/tex]Constant: DNE.
2) Convert this fraction into a mixed number in lowest terms 60/25
Please I need step by step explanation.
The given fraction 60/25 into a mixed number would be 2 2/5 in the lowest terms.
What is the fraction?A fraction is defined as a numerical representation of a part of a whole that represents a rational number.
First, we have to find the whole number,
Determine how many times the denominator enters the numerator. Divide 60 by 25 and preserve just the numbers to the left of the decimal point:
⇒ 60 / 25 = 2.4000 = 2
Then, find a new numerator :
Multiply the result by the denominator and subtract it from the original numerator.
⇒ 60 - (25 x 2) = 10
Now, put together
To obtain this, keep the original denominator and use the solutions from Steps 1 and 2:
⇒ 2 10/25 ⇒ 2 2/5
Thus, the given fraction 60/25 into a mixed number would be 2 2/5 in the lowest terms.
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