The equation of line passing that has a slope of One-half and passes through point (2, –3) is y = 0.5x - 4
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Equations are classified based on degree (value of highest exponents) as linear, quadratic, cubic and so on. Variables can be dependent or independent. Dependent variables depend on other variable while an independent variable do not depend.
The standard form for linear equation is:
y = mx + b
Where m is the slope and b is the y intercept
The equation of line passing that has a slope of One-half and passes through point (2, –3) is:
y - (-3) = (1/2)(x - 2)
y + 3 = 0.5x - 1
y = 0.5x - 4
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Match the polar equations to their rectangular forms-
Applying the rules for the polar equations, the correct pairs are given in the image at the end of the answer.
What are polar equations?
The most well known polar equation rules are given as follows:
For the first pair, when x = 2, we have that:
rcos(θ) = 2.
r = 2/cos(θ)
r = 2 sec(θ), as one divided by the cosine is the secant.
For the second pair, we have that:
x² + y² = 36.
x² + y² = r², hence:
r² = 36
r = 6.
For the third pair, we have that:
x² + y² = 2y
r² = 2rsin(θ).
r²/r = 2sin(θ).
r = 2sin(θ).
For the fourth pair, we have that:
[tex]x = \sqrt{3}y[/tex]
[tex]\frac{y}{x} = \frac{1}{\sqrt{3}}[/tex]
[tex]\frac{y}{x} = \frac{\sqrt{3}}{3}[/tex]
Then the angle theta is given as follows:
θ = arctan(sqrt(3)/3) = pi/6.
For the fifth pair, we have an angle in which the sine and the cosine are equal, hence this angle is of 45º = pi/4.
The image at the end of the answer shows all the matched pairs.
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In geometry,you can use deductive rules to
A
b
c
d
In geometry, you can use deductive rules to prove conjectures.
What is geometry?Geometry is defined as a subfield of mathematics that examines the measurement and relationships of surfaces, solids, solid surfaces, angles, and points. The calculation of a triangle's angles is a geometry example.
Geometry will categorize, quantify, and lack a counterexample. Once a concept has been defined, it can be utilized in subsequent definitions; for instance, parallel lines can be used in the definition of a parallelogram once they have been defined. classification, counterexample, quantification, and parallelogram.
Given Data
In geometry, you can use deductive rules to
In geometry, you can use deductive rules to prove conjectures.
In geometry, deductive rules can be used to demonstrate whether a particular assertion or conjecture is true or untrue. We narrow down a result proving a statement must be true or untrue by using deductive principles, such as definitions, postulates, and other theorems, in a logical order.
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Mai, Bill, and Dante have a total of $105 in their wallets. Dante has 3 times what Mai has. Bill has $10 more than Mai. How much does each have?
Answer:
Mai 19
Bill 29
Dante 57
Step-by-step explanation:
assume
mai = X
Bill = Y
Dante = Z
X + Y + Z = 105
Z = 3X
Y = 10 + X
then
X + Y + Z = X + 3X + 10 + X
5X + 10 = 105
5X = 95
X = 19
Y = 29
Z = 57
Find the length of the arc on a circle with a radius of 2.4 kilometers and is intercepted by a central angle measuring 150°. Leave your answer in terms of π. A. 6π/5kmB. 6 kmC. 12π/5kmD. 2π km
Given:
The radius of the circle = 2.4 kilometers
The measure of the arc = 150°
The length of the arc is given by the formula:
[tex]A=\theta\cdot r[/tex]Note: we will convert the angle from degrees to radian
So, the length of the arc =
[tex](150\cdot\frac{\pi}{180})\cdot2.4=\frac{150}{180}\cdot2.4\cdot\pi=2\pi[/tex]So, the answer will be option D. 2π km
(b) y = x²+2x9 dx = 11x¹⁰ + 10x¹, where 10 dy is the derived function. dx Find the value of p and the value of q.
The expression y=[tex]x^{p} +2x^{q}[/tex]. [tex]\frac{dy}{dx}=11x^{10}+10x^{4}[/tex], where dy/dx is derived function. The p and q values are 11 and 5.
Given that,
The expression y=[tex]x^{p} +2x^{q}[/tex]
[tex]\frac{dy}{dx}=11x^{10}+10x^{4}[/tex], where dy/dx is derived function.
Derived function means for any value of the independent variable, the value of another function is the instantaneous rate of change of the specified function at that value of the independent variable.
We have to find the p and q values.
Take expression
y=[tex]x^{p} +2x^{q}[/tex]
[tex]\frac{dy}{dx}=px^{p-1}+2qx^{q-1}[/tex]
[tex]11x^{10}+10x^{4}=px^{p-1}+2qx^{q-1}[/tex]
Here,
we can see p=11 and
2q=10
q=5
Therefore, the p and q values are 11 and 5.
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PLEASE HELP- Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 25 gallons of fuel, the airplane weighs 2060 pounds. When carrying 45 gallons of fuel, it weights 2188 pounds. How much does the airplane weigh if it is carrying 60 gallons of fuel?
The linear function which can be used to represent the situation is y = 6.4x + 1900 and airplane weight if it is carrying 60 gallons of fuel is 2,284 pounds
How to find linear functiony = mx + b
where,
x = the number of gallons and y = the weight of the aircraft in pounds.Now, we can use the two sets of data points given to us to first solve for the slope m as:
m = (2188 - 2060) / (45 - 25)
= 128 / 20
= 6.4
y - yo = m(x - xo)
y - 2060 = 6.4(x - 25)
y - 2060 = 6.4x - 160
y = 6.4x - 160 + 2060
y = 6.4x + 1900
if it is carrying 60 gallons of fuelx = 60
So,
y = 6.4x + 1900
y = 6.4(60) + 1900
y = 384 + 1900
y = 2,284 pounds
Therefore, the weight of the airplane if it is carrying 60 gallons of fuel is 2,284 pounds
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The quantity of six and a number, times
five
A florist sells a dozen roses for $35.40. She sells individual roses for the same unit cost. How much would 18 roses cost?
Answer: 53.1
Step-by-step explanation: Divide 35.40 by 2 to get 17.7. now add 35.40 with 17.7 and you get 53.1
[tex] \rm \int_0^ \infty \frac{1}{(1 + {x}^{ \phi} {)}^{ \phi} } dx \\ [/tex]
Recall that [tex]\phi = 1 + \frac1\phi[/tex].
In the integral, substitute [tex]y=1+x^\phi[/tex], so that [tex]x=(y-1)^{1/\phi} = (y-1)^{\phi-1}[/tex] and [tex]dx = (\phi-1) (y-1)^{\phi-2} \, dy[/tex]. We have
[tex]\displaystyle I = \int_0^\infty \frac{dx}{(1+x^\phi)^\phi} \\\\ ~~~~ = \frac1\phi \int_1^\infty \frac{(y-1)^{\phi-2}}{y^\phi} \, dy[/tex]
Substitute [tex]y=\frac1z[/tex] and [tex]dy=-\frac{dz}{z^2}[/tex] to transform [tex]I[/tex] to
[tex]\displaystyle I = \frac1\phi \int_0^1 \left(\frac1z-1\right)^{\phi-2} z^{\phi-2} \, dz \\\\ ~~~~ = \frac1\phi \int_0^1 (1-z)^{\phi-2} \, dz[/tex]
Finally, substitute [tex]u=1-z[/tex].
[tex]\displaystyle I = \frac1\phi \int_0^1 u^{\phi-2} \, du \\\\ ~~~~ = \frac1\phi \cdot \frac1{\phi-1} = \frac{\phi-1}{\phi-1} = \boxed{1}[/tex]
The analysis of a data set last year produced a line of best fit with equation y = 2.108x - 0.3818. Data was collected from the same sample this year and produced a line of best fit with equation y= 2.107x + 1.7455. Explain the verticalchange in going from last year's data to this year's data.
The slope of the line has increased, meaning it has become a positive number. This shift means that this year the data points were placed higher than the previous year.
I need help with this problem
We have the following function:
[tex]H(t)=10\sin (2\pi(t-\frac{1}{4}))+15[/tex]a) The initial height occurs at t=0. By substituting this value into the function, we get
[tex]H(0)=10\sin (2\pi(0-\frac{1}{4}))+15[/tex]which gives
[tex]\begin{gathered} H(0)=10\sin (-2\pi\frac{1}{4})+15 \\ H(0)=10\sin (-\frac{2\pi}{4})+15 \\ H(0)=10\sin (-\frac{\pi}{2})+15 \end{gathered}[/tex]since sin(-Pi/2) is equal to -1, we have
[tex]H(0)=-10+15[/tex]Then, H(0)= 5, so the height is 5 feets.
b) The car will make a full rotation when the argument is shifted 2Pi:
[tex]H(t)=10\sin (2\pi(t-\frac{1}{4})+2\pi)+15[/tex]at this point H(t) must be equal to 5 feets. Then, we have
[tex]5=10\sin (2\pi(t-\frac{1}{4})-2\pi)+15[/tex]and we must find t. If we move +15 to the left hand side, we obtain
[tex]\begin{gathered} 5-15=10\sin (2\pi(t-\frac{1}{4})-2\pi) \\ \text{then} \\ 10\sin (2\pi(t-\frac{1}{4})-2\pi)=-10 \end{gathered}[/tex]By moving the coefficient 10 the the right hand side, we get
[tex]\begin{gathered} \sin (2\pi(t-\frac{1}{4})-2\pi)=-\frac{10}{10} \\ \sin (2\pi(t-\frac{1}{4})-2\pi)=-1 \end{gathered}[/tex]we can note that when
[tex]\sin \theta=-1\Rightarrow\theta=-90=-\frac{\pi}{2}[/tex]this implies that the argument of our last result is
[tex]2\pi(t-\frac{1}{4})-2\pi=-\frac{\pi}{2}[/tex]By moving 2Pi to the right hand side, we have
[tex]\begin{gathered} 2\pi(t-\frac{1}{4})=-\frac{\pi}{2}+2\pi \\ 2\pi(t-\frac{1}{4})=\frac{3\pi}{2} \end{gathered}[/tex]Now, by moving 2Pi to the right hand side, we have
[tex]t-\frac{1}{4}=-\frac{\frac{3\pi}{2}}{2\pi}[/tex]which gives
[tex]t-\frac{1}{4}=\frac{3}{4}[/tex]so, t is given by
[tex]\begin{gathered} t=\frac{3}{4}+\frac{1}{4} \\ t=1 \end{gathered}[/tex]That is, n 1 minute, the car will take one full rotation.
c) The maximum height of the car ocurrs at t=0.5 min because in one minute it take one full rotation. Then, at t=1/2 we get
[tex]H(\frac{1}{2})=10\sin (2\pi(\frac{1}{2}-\frac{1}{4}))+15[/tex]which gives
[tex]\begin{gathered} H(\frac{1}{2})=10\sin (2\pi(\frac{1}{4}))+15 \\ H(\frac{1}{2})=10\sin (\frac{\pi}{2})+15 \\ H(\frac{1}{2})=10(1)+15 \\ H(\frac{1}{2})=25 \end{gathered}[/tex]that is, the maximum height is 25 feets
Please help with the following question:
Using it's concept, the probabilities are given as follows:
a) Both go off on schedule: 0.30 = 30%.
b) Both go off on schedule: 0.2 = 20%.
What is a probability?A probability is calculated as the number of desired outcomes in the experiment divided by the number of total outcomes in the experiment.
From the text of the problem, the probability of A and not B is of 0.2, that is:
A x (1 - B) = 0.2.
We can multiply the probabilities as the events are independent.
In item a, the probability of neither is of 0.2, hence:
(1 - A) x (1 - B) = 0.2.
From the first equation, we have that:
A = 0.2/(1 - B).
From the second equation, we have that:
(1 - A) = 0.2/(1 - B).
Hence:
A = 1 - A
2A = 1
A = 0.5.
Then we can find B as follows:
(1 - A) x (1 - B) = 0.2.
1 - B = 0.2/0.5
1 - B = 0.4
B = 0.6.
The probability of both is:
A x B = 0.5 x 0.6 = 0.30 = 30%.
For item b, we consider that:
(1 - A) x B = 0.3.
We also have that:
A x (1 - B) = 0.2.
From the first equation:
B = 0.3/(1 - A).
Then, replacing on the second:
A x (1 - 0.3/(1 - A)) = 0.2
Which has a solution of:
A = 0.5, B = 0.4.
Hence the probability of both is:
0.4 x 0.5 = 0.2 = 20%.
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Isabelle is collecting pledges for a walk-a-thon. Her mother has pledged a flat donation of $9, and her grandmother has pledged $1 per mile. If Isabelle walks a certain distance, the two donors will end up owing the same amount. How much will each donor owe?
Answer:
9 mileds
Step-by-step explanation:
Hi, I need to send you the question as I have written
recall the formula for velocity, which involves distance and time:
velocity = distance / time
since the woman's velocity is: 4.5 Km/h, then in order to cover 10 km, we need to find the time in the equation:
4.5 km/h = 10 km / t
multiply both sides by the time (t)
4.5 * t = 10
divide both sides by 4.5 in order to isolate t completely
t = 10 / 4.5
then
t = 2 hours and 0.2222 hours which can be translated into minutes by multiplying this by 60: 40/3 = 13 minutes and 0.33333 minutes. which can be converted into seconds by multiplying it by 60: 20 seconds
Then she will cover that distance in 2 hours, 13 minutes and 20 seconds.
Hi, can you help me to solve this exercise please. If MK=6 ft, find the length “MKL to the nearest Tenth!
To find the arc length we need to know the angle subtended by the arc.
From the diagram we notice that angle MNK is of 180°.
We also notice that:
[tex]\begin{gathered} m\angle JNK+m\angle KNL=90 \\ 52+m\angle KNL=90 \\ m\angle KNL=90-52 \\ m\angle KNL=38 \end{gathered}[/tex]Then the angle subtended by the arc MKL is 218°.
Now that we know this we need to remember that the arc length is given by:
[tex]s=2\pi r(\frac{\theta}{360})[/tex]in this case the radius is 3 ft (The radius is half the diameter) then we have that:
[tex]\begin{gathered} s=2\pi(3)(\frac{218}{360}) \\ s=11.4 \end{gathered}[/tex]Therefore the arc length is 11.4 ft.
Many people use mobile apps and online software to manage and create personalized budgets. Suppose your task is to develop an online tool for personal budgeting. What information would be included or entered? What types of features would the tool have?
The information that would be included or entered into the application or online tool for personal budgeting should include but not be limited to:
ExpensesExpense CategoriesAuto Expense summationExpense forecastingExpense reporting etc.IncomeIncome categoriesAuto Income summationIncome forecastingIncome reporting etc.The features that the tool will have are:
Item-by-item analysisCalendar Reporting AnalysisEmail and Password for login and User RegistrationCloud functionality so that you are able to back up and retrieve information.Learn more about Budgeting:
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Your teacher wants to put a garden in her backyard with a fence around it. The width needs to be the same size as the shed which is 8ft. If she bought 50 feet of fence and wants the largest garden she can make, what should the length of the garden be? Justify your thinking.
Th length of the garden when the teacher bought 50 feet of fence is 17 feet.
How to calculate the value?From the information, the width needs to be the same size as the shed which is 8ft and she bought 50 feet of fence and wants the largest garden she can make,
It should be noted that perimeter is calculated as:
= 2(Length + Width)
In this case, the perimeter is 50 feet.
2(Length + Width) = 50
2(Length + 8) = 50
2L + 16 = 50
2L = 50 - 16
2L =34
Divide
L = 34/3
L = 17 feet.
The length is 17 feet.
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The position, d, in metres of an object
in motion as it travels is given by:
d= (t + 2) (t + 1.2)
It is the time in seconds.
This is an equation of motion.
By looking at it, we can learn about
how the object started moving.
First, expand the brackets.
t2+2.4+3.2t
The initial height is the constant
(no t) term.
What is the initial height?
Answer:
2.4m
Step-by-step explanation:
The question has already done most of the work for you. To find the initial height, simply set t = 0 and substitute it into the function.
[tex]d(0) = {0}^{2} + 2.4 + 3.2(0) \\ = 2.4m[/tex]
Mary has 65 roses and 104 tulips. What is
the greatest number of bouquets they can be
divided into if there will be the same number
of roses in each bouquet and the same
number of tulips in each bouquet?
The greatest number of bouquets they can be divided into if there will be the same number of roses in each bouquet and the same number of tulips in each bouquet is 5.
Given that:-
Number of roses Mary has = 65
Number of tulips Mary has = 104
We have to find the greatest number of bouquets they can be divided into if there will be the same number of roses in each bouquet and the same number of tulips in each bouquet.
HCF (65, 104) = 13
Hence, the groups of 13 roses and 13 tulips will together make a bouquet.
From 65 roses, 65/13 = 5 bouquets can be made.
From 104 tulips, 104/13 = 8 bouquets can be made.
Min (5,8) = 5.
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7. Write a biconditional for the following conditional. Determine the truth value
of the new statement. If two lines have the same slope, then they are parallel.
Line AB is parallel to CD, for instance, as indicated by the formula AB ll CD.
How do parallel lines work?
Lines in a plane that are regularly spaced apart are known as parallel lines. Parallel lines don't overlap each other. Perpendicular lines are those that cross at a straight angle of 90 degrees.Only if the slopes of two lines are equal can they be said to be parallel. If two lines have different y-intercepts and equal slopes, they are said to be parallel. In a plane, parallel lines are two lines that never cross (meaning they will continue on forever without ever touching). The equal slopes of parallel lines are a distinctive feature. The rise (change in Y coordinates) over the run (change in X coordinates) of a line, or its steepness, is referred to as the slope of a line. The most typical way to depict parallel lines is with two vertical lines (ll). Line AB is parallel to CD, for instance, as indicated by the formula AB ll CD.Learn more about parallel lines
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Find k with the given information.
m=-3/5 (k,-9) and (1,-6
Using the slope formula, the value of k in the coordinates of the point given is: k = 6.
How to Find the Slope of a Line?If we are given two points that lie on a line, (x1, y1) and (x2, y2), the formula that is used to find the slope (m) of the line is given as:
Slope (m) = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex] = change in y / change in x..
Given the following:
(k, -9) = (x1, y1)
(1, -6) = (x2, y2)
Slope (m) = -3/5
Plug in the values into the formula used for calculating the slope of a line:
(-6 -(-9))/(1 - k) = -3/5
3/(1 - k) = -3/5
Cross multiply and solve for the value of k
-3(1 - k) = 3(5)
-3 + 3k = 15
Add 3 to both sides
-3 + 3k + 3 = 15 + 3 [subtraction property of equality]
3k = 18
Divide both sides by 3
3k/3 = 18/3 [division property of equality]
k = 6
The value of k is: 6. Therefore, the points on the line that have a slope of -3/5 are (6, -9) and (1, -6).
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)) 24 cups 5) 3 cups
Answer: 12 cups
Explanation
Anna needs 6 pint of milk to produce yoghurt
1 pint = 2 cups
let cups of milk be x
1 pint of milk will give 2 cups
6 pints of milk will give x cups of milk
1 pint = 2cups
6 pints = x cups
Cross multiply
1 * x = 2 * 6
x = 12 cups
Therefore, Anna will need 12 cups of milk to produce a yoghurt.
While on a holiday in Montego Bay, Jamaica, Marilyn shops for souvenirs for friends. She
has budgeted C$120 for souvenirs.
a) How much does she have in Jamaican dollars to spend on souvenirs?
She spend J$10039.48 in Jamaican currency on souvenirs.
What is currency?The term "money" in mathematics refers to a form of payment, such as bills, coins, and demand deposits, that is used to purchase goods and services. Money is used to pay for the worth or price of an item or service. The US dollar is the country's recognized unit of exchange. A currency is the standardization of money, including banknotes and coins, that is used or in circulation as a means of exchange. A currency can also be defined more broadly as a system of money that has been used often over time within a particular setting, particularly by residents of a country state.
Given Data
While on a holiday in Montego Bay, Jamaica, Marilyn shops for souvenirs for friends. She has budgeted C$120 for souvenirs.
She spend a certain amount in Jamaican currency is:
CD$120(83.6623)
J$10039.48
She spend J$10039.48 in Jamaican currency on souvenirs.
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The set of all numbers greater than or equal to -10 and less than 9
The The set of all numbers greater than or equal to -10 and less than 9 will be -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, and 8.
How to illustrate the information?It should be noted that from the information, we are to find the set of all numbers greater than or equal to -10 and less than 9.
This simply means the numbers from -10 to 8.
Therefore, the numbers will be -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, and 8.
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Can someone help with this question?
The slope of the line is 2 / 3.
How to find the slope of a line?The slope of a line can be found as follows:
The slope of a line is change in y with respect to change in x.
In other words, the slope of a line is rise over run.
Therefore,
slope = y₂ - y₁ / x₂ - x₁
Using (0, 3)(-3, 1)
Hence,
x₁ = 0
x₂ = -3
y₁ = 3
y₂ = 1
Therefore,
slope = 1 - 3 / -3 - 0
slope = -2 / -3
slope = 2 / 3
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To pay for a $22,800car, Mai made a down payment of $3,800and took out a loan for the rest. On the loan, she paid monthly payments of $420for 4 years. (a) What was the total amount Mai ended up paying for the car (including the down payment and monthly payments)? (b) How much interest did Mai pay on the loan?
1. The total amount that was paid for the car will be $23960.
2. The interest that Mai pay on the loan will be $1160.
How to calculate the value?1. From the information, in order to pay for a $22,800 car, Mai made a down payment of $3,800 and took out a loan for the rest and on the loan, she paid monthly payments of $420for 4 years.
The total amount that was paid for the car will be:
= $3800 + ($420 × 4 years)
= $3800 + ($420 × 48)
= $3800 + $20160
= $23960
2. The interest that Mai pay on the loan will be:
= $23960 - $22800
= $1160
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10 less than the quotient of a number and 2 is -18"
The representation of the expression is x/2 - 10 = -18
How to translate the algebraic expression?The mathematical statement is given as
"10 less than the quotient of a number and 2 is -18"
In expressions, quotient mean divide i.e. /
So, we have the following representation
"10 less than the quotient of a number and 2 is -18" ⇒ 10 less than a number/2 is -18"
Let the number be x
So, we have
"10 less than the quotient of a number and 2 is -18" ⇒ 10 less than x/2 is -18"
Less than as used here means minus
So, we have
"10 less than the quotient of a number and 2 is -18" ⇒ x/2 - 10 is -18
Rewrite as
"10 less than the quotient of a number and 2 is -18" ⇒ x/2 - 10 = -18
Hence, the expression represented by "10 less than the quotient of a number and 2 is -18" is x/2 - 10 = -18
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Can the angle measures of a triangle be 46°, 82°and 50°
Answer:Explanation:
No
The sum of the interior angles of a triangle is always 180°. in this case, the sum is:
Then,
46 + 82 + 50 = 178
Since 178 and 180 are distinct, 46°, 82°and 50° can't be the angle measures of a triangle.
Select the correct answer. Based on these segment lengths, which group of segments can form a triangle? A. 3, 10, 14 B. 8, 7, 13 C. 3, 2, 5 D. 20, 7, 13
Answer:
B
Step-by-step explanation:
By the triangle inequality theorem, the sum of the lengths of the two shorter sides needs to be greater than the length of the longest side.
Convert the following phrase into a mathematical expression use x as the variable combine like terms when possible A number multiplied by -8 subtracted from the sum of 8 and two times the number
ANSWER:
[tex]-8x-(8+2x)[/tex]STEP-BY-STEP EXPLANATION:
We must convert the following statement "A number multiplied by -8 subtracted from the sum of 8 and two times the number" into a mathematical expression.
We know that "a number" would be x.
Therefore:
[tex]-8x-(8+2x)[/tex]