Answers
Let's call the y intercept b for now
m = -1/4 is the perpendicular slope
(x,y) = (16,1) is the point the perpendicular line goes through
breaking this down further means x = 16 and y = 1 pair up together
y = mx+b
y = (-1/4)*x+b ... replace m with -1/4
y = (-1/4)*16+b .... replace x with 16
1 = (-1/4)*16+b .... replace y with 1
1 = -4+b ... multiply -1/4 and 16 to get -4
1+4 = -4+b+4 ... add 4 to both sides
5 = b
b = 5
So the y intercept is 5 (not 4)
since m = -1/4 and b = 5 we go from
y = mx+b
to
y = (-1/4)x+5
So the final answer is y = (-1/4)x+5 which can be written as [tex]y = -\frac{1}{4}x+5[/tex]
As a check, we can plug x = 16 into the equation to get...
y = (-1/4)*x + 5
y = (-1/4)*16 + 5
y = -4 + 5
y = 1
so that confirms we have the right answer.
Related Questions
Is this equation correct? 18/48=27/72?
Answers
3x+y=5
y=2x
Substitution method
Answers
3x + y = 5
3x + 2x = 5
5x = 5
x = 1
Now, substitue 1 as x into one of the two equations:
y = 2x
y = 2(1)
y = 2
Hope this helps!
Suppose that f(x+h)−f(x)= −6hx2−7hx−6h2x−7h2+2h3.
Find f′(x).
f′(x)=
Answers
f'(x)=Lim h->0 (f(x+h)-f(x))/h
We are already given
f(x+h)-f(x)=−6hx2−7hx−6h2x−7h2+2h3=h(-6x^2-7x-6hx-7h+2h^2)
divide by h
(f(x+h)-f(x))/h =h(-6x^2-7x-6hx-7h+2h^2)/h=(-6x^2-7x-6hx-7h+2h^2)
Finally, take lim h->0
f'(x)=Lim h->0 (f(x+h)-f(x))/h=(-6x^2-7x-0-0+0)=-6x^2-7x
=>
f'(x)=-6x^2-7x
Final answer:
To find f'(x), simplify the given expression by canceling the common factor of h, then take the limit as h approaches zero. The resulting derivative is f'(x) = -6x² -7x.
Explanation:
To find the derivative of function f(x), denoted as f'(x), we use the definition of the derivative and the given equation f(x+h) - f(x) = -6hx2 - 7hx - 6h2x - 7h2 + 2h3. The definition of the derivative is the limit of the average rate of change as the interval approaches zero, formally:
f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h
In the given difference quotient, every term includes an h, allowing us to simplify by cancelling an h from each term before taking the limit as h approaches zero. This gives us:
f'(x) = lim (h -> 0) [-6x² - 7x - 6h2 - 7h + 2h2]
When h approaches zero, the terms containing h also approach zero. Thus, we are left with:
f'(x) = -6x² - 7x
Which are properties of rotations? Check all that apply.
1 Rotations are always counterclockwise.
2 Rotations preserve size.
3 Rotations of 360° map a figure to itself.
4 Lines connecting corresponding points on the pre-image and image are parallel.
5 Lines connecting corresponding points on the pre-image and image have equal length.
6 Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length.
Answers
Answer:- The correct properties of rotation are as follows:-
2.Rotations preserve size.
3. Rotations of 360° map a figure to itself.
6. Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length.
Explanation:-
Rotation transformation is a rigid transformation which create the image of a figure with same shape and size by rotating it by some degrees about the the center of rotation.1. False ; Rotations are always counterclockwise.→ it is not a property of rotation, ∵rotation can be clockwise or counterclockwise both.
2.True; Rotations preserve size. [by definition of rotation]
3.True; Rotations of 360° map a figure to itself. [It completes one turn and get back to the original position ]
4.False; Lines connecting corresponding points on the pre-image and image are parallel. [They can be intersect]
5.False ; Lines connecting corresponding points on the pre-image and image have equal length.[The point which is the center of rotation in a figure is nearest to its image rather than the other points. ]
6. True ; Lines connecting the center of rotation to the pre-image and the corresponding point on the image have equal length. [∵ all the points of the figure is rotated about the center of the point ,so the distance of the center of rotation to the pre-image and the corresponding point on the image have equal length]
Rotations have several properties, including preserving size, mapping a figure to itself when rotated by 360°, and having lines connecting the centre of rotation to the pre-image and image equal in length.
Properties of rotations:
Rotations are not always counterclockwise.Rotations preserve size.Rotations of 360° map a figure to itself.Lines connecting corresponding points on the pre-image and image are parallel.Lines connecting corresponding points on the preimage and image have equal lengths.Lines connecting the centre of rotation to the pre-image and the corresponding point on the image have equal lengths.How much force is required to push a 54-pound sofa across a carpeted floor?
Answers
Find the minimum and maximum of f(x,y,z)=x^2+y^2+z^2 subject to two constraints, x+2y+z=4 and x-y=8.
Answers
[tex]L(x,y,z,\lambda_1,\lambda_2)=x^2+y^2+z^2+\lambda_1(x+2y+z-4)+\lambda_2(x-y-8)[/tex]
which has partial derivatives (set equal to 0) satisfying
[tex]\begin{cases}L_x=2x+\lambda_1+\lambda_2=0\\L_y=2y+2\lambda_1-\lambda_2=0\\L_z=2z+\lambda_1=0\\L_{\lambda_1}=x+2y+z-4=0\\L_{\lambda_2}=x-y-8=0\end{cases}[/tex]
This is a fairly standard linear system. Solving yields Lagrange multipliers of [tex]\lambda_1=-\dfrac{32}{11}[/tex] and [tex]\lambda_2=-\dfrac{104}{11}[/tex], and at the same time we find only one critical point at [tex](x,y,z)=\left(\dfrac{68}{11},-\dfrac{20}{11},\dfrac{16}{11}\right)[/tex].
Check the Hessian for [tex]f(x,y,z)[/tex], given by
[tex]\mathbf H(x,y,z)=\begin{bmatrix}f_{xx}&f_{xy}&f_{xz}\\f_{yx}&f_{yy}&f_{yz}\\f_{zx}&f_{zy}&f_{zz}\end{bmatrix}=\begin{bmatrix}=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}[/tex]
[tex]\mathbf H[/tex] is positive definite, since [tex]\mathbf v^\top\mathbf{Hv}>0[/tex] for any vector [tex]\mathbf v=\begin{bmatrix}x&y&z\end{bmatrix}^\top[/tex], which means [tex]f(x,y,z)=x^2+y^2+z^2[/tex] attains a minimum value of [tex]\dfrac{480}{11}[/tex] at [tex]\left(\dfrac{68}{11},-\dfrac{20}{11},\dfrac{16}{11}\right)[/tex]. There is no maximum over the given constraints.
A bathtub is draining at a constant rate. After 2 minutes, it holds 30 gallons of water. Three minutes later, it holds 12 gallons of water. Write an equation that represents the number y of gallons of water in the tub after x minutes.
Answers
3 minutes - 2 minutes = 1 minute
30 gallons - 12 gallons = 18 gallons
it is draining at 18 gallons per minute
equation would be: y = -18x
Michael started a savings account with $300. After 4 weeks, he had $350 dollars, and after 9 weeks, he had $400. What is the rate of change of money in his savings account per week?
Answers
Finley makes 1 batch of her favorite shade of orange paint by mixing 5 liters of yellow paint with 3 liters of red paint. How many batches of orange paint can Finley make if she has 15 liters of red paint?
Answers
Jessica purchased an $84 suit. The sales tax is 6.5%. What is the total price of the suit including tax?
Answers
Hope it helped!
What is the logarithmic form of 144=12^2
Answers
The logarithmic form of 144=12^2 is log12(144) = 2.
Explanation:The logarithmic form of 144=12^2 is log12(144) = 2.
The logarithm is the power to which a base number must be raised to equal a given number. In this case, the base is 12, and we are trying to find the power that results in 144.
Since 12^2 = 144, the logarithmic form is log12(144) = 2.
the logarithmic form of 144=12^2 is b: [tex]\( \log_{12}(144) = 2 \)[/tex].
The correct option is (b).
The logarithmic form of the expression [tex]\( 12^2 = 144 \)[/tex] translates an exponential equation into a logarithmic equation. The general form of converting from exponential to logarithmic form is:
[tex]\[ b^y = x \] is equivalent to[/tex]
[tex]\\ \[ \log_b(x) = y \][/tex]
where:
- [tex]\( b \)[/tex] is the base of the exponential expression,
-[tex]\( y \)[/tex] is the exponent,
-[tex]\( x \)[/tex] is the result of the exponential expression.
Applying this to the given equation[tex]\( 12^2 = 144 \)[/tex], we have:
-[tex]\( b = 12 \)[/tex],
- [tex]\( y = 2 \)[/tex],
- [tex]\( x = 144 \)[/tex].
So, the logarithmic form is:
[tex]\[ \log_{12}(144) = 2 \][/tex]
complete question given below:
What is the logarithmic form of 12² = 144?
a.log212=144
b.log12144=2
c.log1442=12
d.log2144=12
G use lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane.x + 9y + 8z = 27
Answers
[tex]L(x,y,z,\lambda)=xyz+\lambda(x+9y+8z-27)[/tex]
with partial derivatives (set equal to 0 to find critical points)
[tex]\begin{cases}L_x=yz+\lambda=0\\L_y=xz+9\lambda=0\\L_z=xy+8\lambda=0\\L_\lambda=x+9y+8z-27=0\end{cases}[/tex]
Solving the first equation for [tex]\lambda[/tex] gives [tex]\lambda=-yz[/tex]. Substituting this into the next two equations, we have
[tex]xz-9yz=0\implies z(x-9y)=0\implies x=9y[/tex]
[tex]xy-8yz=0\implies y(x-8z)=0\implies x=8z[/tex]
Now
[tex]x+9y+8z=27\implies x+x+x=3x=27\implies x=9[/tex]
[tex]x=9y\implies y=1[/tex]
[tex]x=8z\implies z=\dfrac98[/tex]
So the vertex of the cuboid in the given plane that maximizes the cuboids volume is [tex](x,y,z)=\left(9,1,\dfrac98\right)[/tex], giving a volume of [tex]\dfrac{81}8[/tex].
An aluminum recycling center pays $0.08 per pound for cans. How much will Mark receive for 21.5 pounds of cans?
Answers
You have 900-grams of an an unknown radioactive substance that has been determined to decay according to D ( t ) = 900 e − 0.002415 ⋅ t D ( t ) = 900 e - 0.002415 ⋅ t where t t is in years. How long before half of the initial amount has decayed?
It will take __ years for half of the initial amount to decay. (Round to 1 decimal place)
Answers
so, how long will it be, for D(t) to turn to 450 grams?
[tex]\bf D(t)=900e^{-0.002415t}\implies 450=900e^{-0.002415t} \\\\\\ \cfrac{450}{900}=e^{-0.002415t}\implies \cfrac{1}{2}=e^{-0.002415t}\\\\ -------------------------------\\\\ \textit{Logarithm Cancellation Rules}\\\\ log_{{ a}}{{ a}}^x\implies x\qquad \qquad {{ a}}^{log_{{ a}}x}=x\\\\ -------------------------------\\\\[/tex]
[tex]\bf log_e\left( \frac{1}{2} \right)=log_e\left( e^{-0.002415t} \right)\implies log_e\left( \frac{1}{2} \right)=-0.002415t \\\\\\ \cfrac{ln\left( \frac{1}{2}\right)}{-0.002415}=t\implies 287.0174661 \approx t[/tex]
What is 0.10221 in scientific notation
Answers
1.0221 x 10^-1
after 75% reduction, you purchase a new clothes dryer on sale for $200. What was the original price?
Answers
Original price was $800 for purchasing a new clothe dryer on sale for $200 after 75% reduction.
Original price is the price offered initially without applying any offer.
Discount offered is the amount of relaxation that is the difference between the marked price and selling price.
Given the following information:
Discount provided= 75%
Purchasing price after discount=200
Let the original price be x
x-75%x=200
[tex]x-\dfrac{75}{100}x=200\\\\100x-75x=200\times 100\\[/tex]
Solving the equation as follows to calculate the value of x:
[tex]25x=20000\\x=20000/25\\x=800[/tex]
Thus, original price was $800.
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clark wants to figure out how many pens to order for the office. there are 36 workers, and he need to order one pen for each worker. he know that for every eight people who prefer red , there are four who prefer blue. How many blue pens should he order?
Answers
8 like red 4 like blue
8+4 = 12
36/12=3
4*3 = 12 blue pens
Evaluate. 4⋅(4+(2^6⋅2^−3))
Answers
First, settle the exponents. 2⁶ x 2⁻³ = 2³ = 8
4⋅(4+8)
Settle within the parentheses
4 (12)
48 is your answer
The table below shows the results of a survey in which 141141 men and 145145 women workers ages 25 to 64 were asked if they have at least one month's income set aside for emergencies. complete parts (a) through (d). men women total less than one month's income 6565 8484 149149 one month's income or more 7676 6161 137137 total 141141 145145 286286 (a) find the probability that a randomly selected worker has one month's income or more set aside for emergencies. the probability is nothing. (round to the nearest thousandth as needed.) (b) given that a randomly selected worker is a male, find the probability that the worker has less than one month's income. the probability is nothing. (round to the nearest thousandth as needed.) (c) given that a randomly selected worker has one month's income or more, find the probability that the worker is a female. the probability is nothing. (round to the nearest thousandth as needed.) (d) are the events "having less than one month's income saved" and "being male" independent or dependent? independent dependent
Answers
Final answer:
The probability that a randomly selected worker has one month's income or more set aside for emergencies is approximately 0.479. Given that a randomly selected worker is a male, the probability that the worker has less than one month's income is approximately 0.460. Given that a randomly selected worker has one month's income or more set aside for emergencies, the probability that the worker is a female is approximately 0.446. The events 'having less than one month's income saved' and 'being male' are independent.
Explanation:
To find the probability that a randomly selected worker has one month's income or more set aside for emergencies, we need to divide the number of workers with one month's income or more by the total number of workers. According to the table, there are 76 men and 61 women with one month's income or more, for a total of 137 workers. The total number of workers is 286. So the probability is 137/286, which is approximately 0.479.
Given that a randomly selected worker is a male, we need to find the probability that the worker has less than one month's income set aside for emergencies. According to the table, there are 65 men with less than one month's income. The total number of men is 141. So the probability is 65/141, which is approximately 0.460.
Given that a randomly selected worker has one month's income or more set aside for emergencies, we need to find the probability that the worker is a female. According to the table, there are 61 women with one month's income or more. The total number of workers with one month's income or more is 137. So the probability is 61/137, which is approximately 0.446.
The events 'having less than one month's income saved' and 'being male' are independent. To confirm independence, we need to check if the probability of both events occurring is equal to the product of their individual probabilities. In this case, the probability of having less than one month's income saved is 65/286 and the probability of being male is 141/286. The probability of both events occurring is (65/286) * (141/286), which is approximately 0.126. This is equal to the product of their individual probabilities, confirming that the events are independent.
How much will you pay the cashier for a candy bar that originally costs 79 cents and is now marked 15% off? Sales tax is 5.5%
Answers
so
79 x (1 - 0.15)
79 x 0.85 = 67 cents
plus Sales tax is 5.5%
67 x (1.055) = 71 cents
answer
71 cents
a plumber charges a base fee of $55 for a service call plus $35 per hour for each hour worked during the service call the relationship between the total price of the service called and the number of hours worked
Answers
The relationship between the total price and the number of hours worked can be represented by a linear equation. The equation is y = 35x + 55, where y represents the total price and x represents the number of hours worked.
Explanation:The relationship between the total price of the service call and the number of hours worked can be represented by a linear equation. We can use the formula y = mx + b, where y represents the total price, x represents the number of hours worked, m represents the hourly fee, and b represents the base fee.
In this case, the plumber charges a base fee of $55 and $35 per hour. So the equation that expresses the total price is y = 35x + 55.
The independent variable in this situation is the number of hours worked, which is x. The dependent variable is the total price, which is y. The y-intercept of the equation is 55, which represents the base fee. The slope of the equation is 35, which represents the hourly fee. The y-intercept (b) is the point where the line intersects the y-axis, and the slope (m) determines how steep the line is.
What is the equation of the line that passes through
(10, 4)
and is perpendicular to
5x−y=3
Answers
5x - y = 3
-y = -5x + 3
y = (-5x + 3)/(-1)
y = 5x - 3
The slope for this equation is 5.
The slope of the equation we want is the negative reciprocal of 5.
Let m = slope of equation we want.
m = -1/5
We are given the point (10,4).
Plug both into the point-slope formula.
y - 4 = (-1/5)(x - 10)
y - 4 = (-1/5)x + 2
y = (-1/5)x + 2 + 4
The equation we want is
y = (-1/5)x + 6
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A restaurant requires customers to pay a 15% tip for the server. your family spends $45 on the meal.
a.how much money do you need to pay the server?
b.how much money will you spend total?
Answers
A) 45 * 0.15 = $6.75 paid to server
B 45 + 6.75 = $51.75 total
What is the relationship between 0.04 and 0.004?
Answers
hope this helps
Hope this helped!
Find each percentage decrease. round to the nearest percentage from 16 bagels to 0 bagels
Answers
To solve this problem:
The formula would be: New Value – Old Value)/Old Value] x 100 or in the form of variable it would be: [(B-A)/A] x 100 = Answer.
So plugging in the information given in our problem it would be:
0 – 16 / 16 x 100 = percent of change
0/16 x 100 = 0%
The answer is 0%
Find dy/dx by implicit differentiation. 3x + y = 8 + x2y2
Answers
dy/dx - 2x²ydy/dx = 2xy² -3
(1 - 2x²y) dy/dx = (2xy²-3)/(1-2x²y)
To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x, apply the chain rule and product rule, isolate the terms involving dy, and simplify the equation to find dy/dx.
Explanation:To find dy/dx by implicit differentiation, we need to differentiate both sides of the equation with respect to x. Using the chain rule and product rule, we can simplify the equation and solve for dy/dx. Starting with 3x + y = 8 + x^2y^2:
Step 1: Differentiate each term with respect to x.
Step 2: Apply the chain rule and product rule to simplify the equation.
Step 3: Solve for dy/dx by isolating the terms involving dy.
Step 4: Simplify the equation and rearrange to get dy/dx.
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T Shirt sore keeps 7 white T-Shirts on the shelves for every 3 purple T- Shirts on the shelve how many white T-Shirts on the shelve if 15 purple T-Shirts on the shelve. Show how you get the answer in long form
Answers
[tex]15 \div 3 = 5 \\ 5 \times 7 = 35[/tex]
THE OTHER WAY I DID IT WAS;
[tex]3 + 3 + 3 + 3 + 3 = 15 [/tex]
I ADDED IT 5 TIMES AND GOT 15
[tex]7 + 7 + 7 + 7 + 7 = 35[/tex]
I ADDED IT 5 TIMES AND GOT 35
write 4.51 in expanded form
Answers
This is equivalent to 4*1 + 5*(1/10) + 1*(1/100).
The expanded form of 4.52 is 4 + 0.5 + 0.01
Given.
4.51
We need to find the expanded form.
What is expanded form?It is a form of writing a given number by splitting it based on the place value.
Example:
56743
We can write it as:
50000 + 6000 + 700 + 40 + 3
Find the expanded form of 4.51
4 is in ones place
So,
4 - 4
5 is in tenths place
So,
0.5 - 0.5
1 is in hundredths place
So,
.01 - .01
The expanded form is:
4 + 0.5 + 0.01
Thus the expanded form of 4.52 is 4 + 0.5 + 0.01
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Manuel and Sonja are shopping for school supplies. Manuel is buying 5 notebooks and 3 pens at a cost of $21. Sonja is buying 6 notebooks and 5 pens at a cost of $28. The first equation, representing Manuel’s purchase, is mc015-1.jpg. What is the second equation needed to solve the system?
Answers
6x + 5y = 28 (or $28).
Answer:
Second equation is 6x + 5y = 28
Step-by-step explanation:
Let x is the cost of notebooks and y is the cost of pens.
Now we know Manuel is buying 5 notebooks and 3 pens at a cost of $21.
To represent this statement equation is 5x + 3y = 21
Similarly we will form the second equation to solve the system of equations.
Cost of 6 notebooks = 6x
Cost of 5 pens = 5y
Total cost of 6 notebooks and 5 pens = $28
So 6x + 5y = 28 will be the second equation.
By solving this system of equations we can find the cost of notebook as well as cost of pen purchased by Manuel.
Find the instantaneous rate of change of w with respect to z if w=7/3z^2
Answers
The instantaneous rate of change of 'w' with respect to 'z', for the function w = 7/3z², is 14/3*z.
Explanation:The question is asking for the instantaneous rate of change of 'w' with respect to 'z'. To compute this, we need to take the derivative of w with respect to z, for the given function w = 7/3z². Using the power rule, which says that the derivative of z^n is n*z^(n-1), our derivative would be: (7/3)*2*z^(2-1) simplifying this gives 14/3*z. Hence, the instantaneous rate of change of 'w' with respect to 'z' for given function is 14/3*z.
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