Mathematics High School

anna owes home antiques Whose value is 160000 today assuming normal growth. however anna believes the value will grow at 16% per year for the next three years she wants to take this rapis growth ibto consideration when valuing the buisness for potential sale find the future value of the buisness in three years then use that future value to find the present value at a rate 8% compounded annually

Answers

Answer 1
The formula is
A=p (1+r)^t
A future value?
P present value 160000
R interest rate 0.16
T time 3years
A=160,000×(1+0.16)^(3)
A=249,743.36

Use that future value to find the present value at a rate 8% compounded annually
To find p (present value) solve the formula for p
P=A÷ (1+r)^t
Where r is 0.08
P=249,743.36÷(1+0.08)^(3)
p=198,254.33

Related Questions

Hillary rolls 2 number cubes numbered 1 through 6 while playing her favorite board game. She will get a second turn if she rolls a sum that is an odd number greater than 10. What are Hillary's chances of getting a second turn when she rolls the number cubes?

Answers

There is only one odd sum greater than 10 for a pair of dice, 11, and there are only two ways to roll that, a 5 and 6 or a 6 and 5.

P(11)=(2/6)(1/6)

P(11)=2/36

P(11)=1/18
The sample space = {36} total outcomes:
 Te odd sum  greater than 10 is:
either 5 + 6 = 11  or 6 + 5 =11

to get 5 & 6, P= 1/36
OR to get 6 & 5, P = 1/36
In "either or" you will add the probabilities"

P(to get a chance of getting a second turn)=1/36+1/36 = 2/36 = 1/18=0.0555

the area of a rectangle with width x and length 7x is 7x^2 what does the coefficient 7 mean in terms of the problem

Answers

Area of figure is the total number of 1 square unit in the figure. For a rectangle, it is calculated from the product of its length and its width. For this case, we are given:

Width = x 
Length = 7x
Area = x(7x) = 7x^2

The coefficient 7 would mean that the length is 7 times longer than the width, x.

Ted and Meg have each drawn a line on the scatter plot shown below:
Which line best represents the line of best fit?

Line P, because it is closest to most data points
Line P, because it shows a positive association
Line R, because it is closest to most data points
Line R, because it shows a negative association

Answers

line P is the best fit, since the data points are closest to it,

 so the first answer.

Answer:

The line which best represents  the line of best fit is:

 Line P, because it is closest to most data points    

Step-by-step explanation:The line of best fit is a line which best represents the data and the data points are closely related to it.It is also known as a trend line.The line of best fit may pass through, some , none or all of the data points.Also if the scatter plot does not pass through all the data point then the magnitude of positive residual must be approximately equal to the magnitude of negative residual of the data points.

Hence, by looking at the scatter plot we see that all the data points lie above Line R.

Hence, the Line R will not be a line of best fit.

But all the data points lie close to the line P , hence it act as a line of best fit.

which of the following are not necessary when proving that the diagonals of a rectangle are congruent? check all that apply

Answers

opposite sides are perpendicular

all obtuse angles are congruent

Answer:

The correct options are B and D.

Step-by-step explanation:

A quadrilateral is called rectangle if the opposite sides are congruent and parallel to each other. All interior angles are right angle and congruent.

To prove that the diagonals of a rectangle are congruent the necessary conditions are

1. Opposite sides of a rectangle are congruent.

2. All right angles are congruent.

Therefore options A and C are necessary conditions. Option A and C are incorrect.

The opposite sides of a rectangle are parallel to each other and two parallel lines never intersect each other.

Therefore the opposite sides are not perpendicular to each other. Option D is correct.

The angle whose measure is more than 90 degree is called an obtuse angle.

Since all interior angles of a rectangle are right angle and congruent, therefore there is no obtuse angle. So, condition B is unnecessary. Option B is correct.

To make bread dough, a baker mixes flour, eggs, yeast and salt by weight in pounds in the ratio of 11:9:3:2, respectively. how many pounds of yeast are required to make 20 pounds of bread dough?

Answers

In a standard batch, you have 11 pounds of flour, 9 pounds of eggs, 3 pounds of yeast and 2 of salt. This means the bread dough will weigh 11+9+3+2=25 pounds. The ratio of yeast to dough is then 3:25. To get 20 pounds of dough, we need x amount of yeast. Since its a ratio, 3/25 = x/20. We then solve for x, so x = 20*(3/25) = 12/5 = 2.4 pounds of yeast.

which residual plot shows that the line of best fit is a good model?

Answers

The residual plot with a line of best fit that is a good model is the third residual plot.

Which line of best fit is a good model?

The line of best fit should cut across data points in such a way that the data points on each side are relatively the same number.

The data points on both sides should also be roughly the same distance away from the line. The third residual plot fits these parameters and so shows the line of best fit as a good model.

Find out more on the Line of Best fit at https://brainly.com/question/21241382.

Answer:

the 3rd one

Step-by-step explanation: took the test on edu

Which is an equation of a circle with center (2, -1) that passes through the point (3, 4)?

1. (x + 2)2 + (y - 1)2 = 26
2. (x + 2)2 + (y - 1)2 = 13
3. (x - 2)2 + (y + 1)2 = 26
4. (x - 2)2 + (y + 1)2 = 13

Answers

The equation for a circle is (x-h)^2 + (y-k)^2 = r^2. Because we have a center and a coordinate, all we have to solve for is the radius.  Fill in the equation like this: (3-2)^2 + (4+1)^2 = r^2.  That gives us 1^2 + 5^2 = r^2 and 26=r^2.  Now we use that squared radius along with the center to complete the equation for the circle: (x-2)^2 + (y+1)^2 = 26 which is #3 above.

The sum of three numbers is 91. the third number is 2 times the first. the first number is 7 less than the second. what are the numbers?

Answers

Let's call the three numbers a, b, and c.
Now we can turn the information we are given into equations.
The sum of the three numbers is 26:
a + b + c = 26
Twice the first (2 times a) minus the second (2 times a minus b) is 2 less than the third:
2a - b = c - 2
The third is the second minus three times the first:
c = b - 3a
Counting what we have here, we now have three equations and three variables: enough to solve the whole system of equations.
The third equation gives us c directly, so we can start there and substitute into the second equation:
2a - b = (b - 3a) - 2
2a + 3a = b + b - 2
5a = 2b - 2
Let's get one of these variables on its own so we can continue with the substitution:
5a + 2 = 2b
b = (5a + 2) / 2
Now we have c in terms of a and b, and b in terms of just a. So let's use the first equation and substitute to find out what a is:
a + b + c = 26
a + (5a + 2) / 2 + (b - 3a) = 26
a + (5/2)a + 1 + (5a + 2) / 2 - 3a = 26
7/2a + 1 + 5/2a + 1 - 3a = 26
12/2a + 2 - 3a = 26
6a - 3a = 26 - 2
3a = 24
a = 8
At last, we have solved for one of the variables. Now, plug this into the equation for b to find b:
b = (5a + 2) / 2 = (5(8) + 2) / 2 = (40 + 2) / 2 = 42 / 2 = 21
Now we have a and b. Time to find c!
a + b + c = 26
(8) + (21) + c = 26
29 + c = 26
c = 26 - 29
c = -3
So our values for a, b, and c are 8, 21, and -3.
Final answer:

To solve for three unknown numbers with given relationships, we set up algebraic equations. Using substitution, we find the three numbers by solving the linear system. The numbers are 21, 28, and 42.

Explanation:

The problem presented is a typical linear equation problem found in mathematics where we need to find the values of three unknown numbers with the given relationships between them. To find the three numbers, we set up algebraic equations based on the information provided. Let's denote the first number as x, the second number as y, and the third number as z.

According to the problem:

The sum of the three numbers is 91: x + y + z = 91.The third number is 2 times the first: z = 2x.The first number is 7 less than the second: x = y - 7.

Let's now use substitution to solve for the numbers. Substituting the value of x from the third equation into the second equation, we get z = 2(y - 7). Plugging the expressions for x and z back into the first equation, we have y - 7 + y + 2(y - 7) = 91. Simplifying this equation gives us 4y - 21 = 91, which solves to y = 28. Using y = 28, we find x = 21 and z = 42.

Therefore, the three numbers are 21, 28, and 42.

The trees at a national park have been increasing in numbers. There were 1,000 trees in the first year that the park started tracking them. Since then, there has been one fifth as many new trees each year. Create the sigma notation showing the infinite growth of the trees and find the sum, if possible.

1 1000
2 200
3 40

Answers

The sequence forms a Geometric sequence as the rule to obtain the value for the next term is by ratio

Term 1: 1000
Term 2: 200
Term 3: 40

From term 1 to term 2, there's a decrease by [tex] \frac{1}{5} [/tex]
From term 2 to term 3, there's a decrease also by [tex] \frac{1}{5} [/tex]

The rule to find the [tex]n^{th} [/tex] term in a sequence is 
[tex]n^{th}=a r^{n-1} [/tex], where [tex]a[/tex] is the first term in the sequence and [tex]r[/tex] is the ratio

So, the formula for the sequence in question is
[tex]n_{th} [/tex] term = [tex]1000( \frac{1}{5} ^{n-1} )[/tex]

The sequence is a divergent one. We can always find the value of the next term by dividing the previous term by 5 and if we do that, the value of the next term will get closer to 'zero' but never actually equal to zero.

We can find a partial sum of the sequence using the formula
[tex]S_{∞} = \frac{a}{1-r} [/tex] for -1<r<1 
Substituting [tex]a=1000[/tex] and [tex]r= \frac{1}{5} [/tex] we have 
[tex] S_{∞} [/tex] = [tex] \frac{1000}{1- \frac{1}{5} } [/tex] = 1250

Hence, the correct option is option number 1

Answer:

A

Step-by-step explanation:

points (0,0) and (3,3) lie on line k what is the slope of the line that is perpendicular to k

Answers

slope equation: y2-y1 over x2-x1 
3-0 over 3-0 = 1 
to get the perpendicular slope of a line, you must use the negative reciprocal therefore, the slope of a line that is perpendicular to k is equal to -1

Find the volume of each figure to the nearest tenth. Show your work.

Answers

a. Sphere.  V = (4/3)πr³    [π≈3.14;  r=6]

V = (4/3) * 3.14 * 6³ ≈ 904.3  m³


b. Сone.   V = (1/3)πr²h   [π≈3.14;  r=5]

By the Pythagorean theorem:
h = √(13²-5²) = √144 = 12 m

V = (1/3) * 3.14 * 5² * 12 ≈ 314  m³


c. Rectangular parallelepiped.   V = lwh  [l=11, w=5,  h=6]

V = 11 * 5 * 6 = 330 cm³

The graph below shows a line segment PQ: graph of line PQ going through ordered pairs negative 1, 3 and 2, negative 3 Which of the following equations best represents the line segment PQ?A: y = −3x + 2B: y = −2x + 1C: y = −3x − 2D: y = −2x − 1

Answers

You actually don't need a graph to be able to solve this equation. As long as you are given two data points, you can already solve for the linear equation.

The most common way of writing linear equations is the slope-intercept form:
y = mx + b, where m is the slope and b is the y-intercept.

The slope is the ratio of the change of y to the change of x coordinates. Let's say point 1 is (-1,3) and point 2 is (2,-3). Then,

m = (-3-3)/(2-⁻1) = -2

Next, we use any of the given point to the standard equation along with the m to find b. Let's use (-1,3)

3 = -2(-1)+b
b = 1

Thus, the complete linear equation is y = -2x + 1. The answer is B.

Answer:

y = -2x + 1. The answer is B.

Step-by-step explanation:

The line through the midpoint of and perpendicular to the segment joining points (1, 0) and (5, -2). The answe is x_ _= something

Answers

First, you have to find the midpoint of the segment. Using the midpoint theorem, (which is (x1+x2)/2, (y1+y2)/2 ) the midpoint of that segment in (3,-1)

Next find the slope of the segment 1,0 and 5,-2  using rise over run or deltax / deltay which is a fancy way of saying change in x over change in y. 

The slope is 4/2 or 2 
When a line is perpendicular to another the slope is the negative reciprocal so the slope of our new line will be -1/2

Now plug in the coordinate (3, -1)  to find the y-intercept and the equation of that line will be complete


so -1 = -1/2 (3) +b
-1= -1.5 +b
0.5 = b


The final equation will be -1/2x + 0.5 =y

hope this helps:)

What is the simplified form of 3 over 4x plus 3 + 21 over 8 x squared minus 14x minus 15

6 times the quantity 2 x plus 5 end quantity over the quantity 2x minus 5 end quantity times 4 x plus 3
6 over the quantity 4 x plus 3
6 times the quantity x plus 1 end quantity over the quantity 2x minus 5 end quantity times 4 x plus 3
6 times the quantity x plus 1 end quantity over the quantity 2x plus 5 end quantity times 4 x plus 3

Answers

First we have to factorize:
8 x² - 14 x - 15 = 8 x² - 20 x + 6 x - 15 =
= 4 x ( 2 x - 5 ) + 3 ( 2 x - 5 ) =
= ( 2 x - 5 ) ( 4 x + 3 )
Then we will put it into the equation:
3 / ( 4 x + 3 ) +  21 / ( 2 x - 5 ) ( 4 x + 3 ) =
= 3 ( 2 x - 5 ) + 21 / ( 2 x - 5 ) ( 4 x + 3 ) =
= 6 x - 15 + 21 / ( 2 x - 5 ) ( 4 x + 3 ) =
= (6 x + 6) / ( 2 x - 5 ) ( 4 x + 3 ) =
= 6 ( x + 1 ) / ( 2 x - 5 ) ( 4 x + 3 )
Answer: C ) 6 times the quantity x plus 1 end over the quantity 2 x  minus 5 end quantity times 4 x plus 3.    

Answer: Third Option is correct.

Explanation:

Since we have given that

[tex]\frac{3}{4x+3}+\frac{21}{8x^2-14x-15}[/tex]

Now, we will simplify it, step by step:

First we take 3 as common factor :

[tex]3[\frac{1}{4x+3}+\frac{7}{8x^2-14x-15}][/tex]

Now, we will do the method " Splitting the middle term" we get,

[tex]3[\frac{1}{4x+3}+\frac{7}{8x^2-20x+6x-15}]\\\\3[\frac{1}{4x+3}+\frac{7}{4x(2x-5)+3(2x-5)}]\\\\3[\frac{1}{4x+3}+\frac{7}{(4x+3)(2x-5)}]\\\\3[\frac{2x-5+7}{(2x-5)(4x+3)}]\\\\=3[\frac{2x+2}{(2x-5)(4x+3)}]\\\\=\frac{6(x+1)}{(2x-5)(4x+3)}[/tex]

Hence, 6 times the quantity x plus 1 end quantity over the quantity 2x minus 5 end quantity times 4x plus 3.

Therefore, Third Option is correct.


The function below shows the number of car owners f(t), in thousands, in a city in different years t: f(t) = 0.25t2 − 0.5t + 3.5 The average rate of change of f(t) from t = 2 to t = 6 is ______ thousand owners per year. Answer for Blank 1:

Answers

We solve this by the definition of slope in analytical geometry. The definition of slope is the rise over run. In equation, that would be

m = Δy/Δx = (y₂-y₁)/(x₂-x₁)

The x-coordinates here are the t values, while the y-coordinates are the f(t) values. So, let's find the y values of the boundaries.

At t=2: f(t)= 0.25(2)² − 0.5(2) + 3.5 = 3.5
Point 1 is (2, 3.5)
At t=6: f(t)= 0.25(6)² − 0.5(6) + 3.5 = 9.5
Point 2 is (6, 9.5)

The slope would then be

m = (9.5-3.5)/(6-2)
m = 1.5

Hence, the slope is 1.5. Interpreting the data, the rate of change between t=2 and t=6 is 1.5 thousands per year.

Describe the graph of the function.y = |x – 4| – 7

Answers

This is always ''interesting'' If you see an absolute value, you always need to deal with when it is zero:

(x-4)=0 ===> x=4,

so that now you have to plot 2 functions!

For x<= 4: what's inside the absolute value (x-4) is negative, right?, then let's make it +, by multiplying by -1:

|x-4| = -(x-4)=4-x
Then:

for x<=4, y = -x+4-7 = -x-3

for x=>4, (x-4) is positive, so no changes:

y= x-4-7 = x-11,

Now plot both lines. Pick up some x that are 4 or less, for y = -x-3, and some points that are 4 or greater, for y=x-11

In fact, only two points are necessary to draw a line, right? So if you want to go full speed, choose:

x=4 and x= 3 for y=-x-3

And just x=5 for y=x-11

The reason is that the absolute value is continuous, so x=4 works for both:

x=4===> y=-4-3 = -7

x==4 ====> y = 4-11=-7!

abs() usually have a cusp int he point where it is =0

Hope it helps, despite being this long!
Answer with explanation:

We are given a function as:

                 [tex]y=|x-4|-7[/tex]      

We know that this function is a transformation of the parent function y=|x| i.e. the  modulus function.

The rule that holds in this transformation is:

It is a translation of the parent function 4 units to the right and 7 units down.

Also, the graph of the function is attached to the answer.

What is the measure of ABC?

Answers

Angle ABC is 65 degrees 
You have to divide 130 by 2 because it is the interior angle of arc AC
130/2=65
hope this helps:)

Answer: Measure of <ABC=65°.

Step-by-step explanation:

The intercepted arc Theorem says :Angle made on circumference if half the measure of the intercepted arc.

In the given circle measure of arc is 130° The angle intercepted by arc AC is <ABC.

m<ABCwill be equal to Half of measure of arc AC

M<ABC=[tex]\frac{1}{2} of 130=65[/tex]

what is the range of f(x)=log0.6x

Answers

The range of a log is all real numbers, only the domain is limited. Therefore the answer is +infinity to -infinity.


A bag contains 21 red marbles, 22 green marbles, 24 orange marbles, and 10 yellow marbles. You choose a marble randomly from the bag.

What is the approximate probability that you will choose a green or yellow marble?

A. 0.358
B. 0.435
C. 0.416
D. 0.538

Answers

21 +22+24+10 = 77 total marbles

22 +10 =32 green and yellow marbles

32/77 probability of choosing a green or yellow one

32/77 = 0.416

 answer is C. 0.416


On Mars, if you hit a baseball, the height of the ball at time "t" would be modelled by the quadratic function h=-1.85t^2+20t+1, where t is in seconds and h is in metres. When will the ball hit the ground?

Answers

Given that the height of the baseball is modeled by the equation:
h(t)=-1.85t^2+20t+1
where t is time in seconds and h is the height in meters
The time taken for the ball to hit the ground will be found as follows;
at the point when the ball hits the ground, h(t)=0, therefore our equation will be:
-1.85t^2+20t+1=0
solving the above quadratic equation for t we get:
t=[-b+/-sqrt(b^2-4ac)]/(2a)
substituting the values into the above formula we get:
t=[-20+\-sqrt(20^2+7.4)]/(-3.7)
t=-0.049771
or
t=10.8606
but since there is no negative time values we shall select:
t=10.8606
hence, we conclude that the time taken for the ball to hit the ground was:
t=10.8606 seconds

A key code must contain 6 numerals. There are 10 numerals available. Using these numerals, how many different key codes may be created? A. 151,200 B. 340 C. 210 D. 3,480

Answers

[tex]10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5=151,200[/tex]

Answer: Option A, 151,200 possible combinations.

Explanation:

The code must contain 6 numbers, and there are a total of 10 numbers avaible. Here we must assume that the numbers must be different, because of the word "avaible" and the options avaible.

This means that for the first number, we have 10 options, for the second number, we have 9 options (because we already took 1), in the third number, we have 8 options, and so on.

The total number of combinations is equal to the product between the number of options for each number in the code.

10*9*8*7*6*5 = 151,200

Then the right option is A.

Two lines are perpendicular. if one line has a slope of 0, what is the slope of the other line?

Answers

If two lines are perpendicular, then one line is an opposite reciprocal of the other. Since one line has a slope of 0, the opposite reciprocal of 0 would be -0/0, but any number over 0 is described as undefined, so the slope of the other line is undefined.

A square with an area of 64 in2 is rotated to form a cylinder. What is the radius of the cylinder?

Answers

The area of the square is calculated through the equation,

                               Area = e²

where e is the measure of sides.
 
From the given,

                              64 = e²
                               e = sqrt 64 = 8 inches

The length of the side of the square rotated to form the cylinder is the diameter of the base of the cylinder. To get the radius, we divide the diameter by 2.

                             radius = 8 inches / 2 = 4 inches

Thus, the radius of the cylinder is equal to 4 inches. 

What is the equation of the blue graph?

Answers

Answer should be D. G(x) = (x - 5)^2


G(x) = (x - 5)^2 ....move F(x) = x^2 to the right 5 units

Hope it helps

Okay before i even ask, everyone should i suck at math. so: Michelle went to Spain for 22 days in June. What fraction of the month June did Michelle spend in Spain?

Answers

Number of days in June = 30

Number of days Michelle spent in Spain in June = 22

Hence, fraction of the month Michelle spent in Spain = [tex] \frac{22}{30} = \frac{11}{15} [/tex]

Find the value of X.

Answers

RS + ST = RT

3x + 1 + 2x - 2 = 64
5x - 1 = 64
5x = 64 + 1
5x = 65
x = 65/5
x = 13 <==

State two reasons why including the​ p-value is prudent when you are reporting the results of a hypothesis test.

Answers

In general as we know when we perform a hypothesis test in statistics then p value help us to determine the significance of our result.
Hypothesis test are generally used to test a validity of claim that is made about a population.This claim that's on trial in essence is called as a Null Hypothesis.

The p - values is a number between 0 to 1 and interpreted in following way:

A small p - value (<= 0.05) indicates the strong evidence against a null hypothesis , so you reject a null hypothesis.

A large p - value (>0.05) indicates weak evidence against a null hypothesis , so you fail to reject the null hypothesis.

The reason which include the p - value is prudent when you are reporting the result of a hypothesis test are given below:

1. It allows us for evaluating the strength of the evidence against the null hypothesis.

2. Allow us for assessing significance at any desired level . The null hypothesis can be rejected at any significance level larger than or equal to p -  value and it can not be rejected at any significance level smaller than the p - value.

A carpet cleaning business has 100 customers when they charge $125 for a cleaning. Research shows that every $5 reduction in price attracts another 20 customers. What price should the business implement to maximize its revenue?
a $75
b $100
c $120
d $90

Answers

Since a $5 decrease in price increases customers by 20 we can say that we have two points:

(125,100) and (120,120), from these we can find the slope or rate of change of customers as a function of price...

m=20/-5

m=-4

m=-4

c(p)=-4p+b, now we can use (125,100) to solve for b

100=-4(125)+b

100=-500+b

600=b, so our number of customers as a function of price is:

c(p)=600-4p

Revenue will simply be the number of customers times the price charged per customer...or p*c(p):

r(p)=600p-4p^2

We can find price that creates maximum revenue by finding when the derivative is equal to zero...

dr/dp=600-8p

dr/dp=0 only when

0=600-8p

8p=600

p=75

So the price that maximizes revenue is $75.

The linear relationship is c(p) = -4p + b. Then the price that maximizes revenue is $75.

What is the linear system?

It is a system of an equation in which the highest power of the variable is always 1.

A carpet cleaning business has 100 customers when they charge $125 for cleaning.

Research shows that every $5 reduction in price attracts another 20 customers.

Since a $5 decrease in price increases customers by 20, we can say that we have two options.

(120, 100) and (120, 120) from these we can find the slope or rate of change of customers as a function of price.

[tex]\rm m = \dfrac{20}{-5}\\\\ m = -4[/tex]

Then we have

[tex]\rm c(p ) = -4 p +b[/tex]

Now, we can use (125, 100) to solve for b

100 = -4(125) + b

   b = 600

So our number of customers as a function of price is

[tex]\rm c(p) = 600 - 4p[/tex]

Revenue will simply be the number of customers times the price charged per customer will be

[tex]\rm r(p) = 600 p -4p^2[/tex]

We can find a price that creates maximum revenue by finding when the derivative is equal to zero.

[tex]\rm \dfrac{dr}{dp} = 600 - 8p\\\\[/tex]

WE know that for maximum,

[tex]\rm \dfrac{dr}{dp} = 0\\\\0 = 600 - 8p\\\\p = 75[/tex]

So, the price that maximizes revenue is $75.

More about the linear system link is given below.

If the tangent line to y = f(x) at (7, 4) passes through the point (0, 3), find f(7) and f '(7).

Answers

hello : 
f'(7) = the slope of the tangent line : let  A(7,4)    B(0,3)
the slope is :   (YB - YA)/(XB -XA)= (3-4)/(0-7) = -1/-7 = 1/7=f'(7)
the equation of the tangent line is :
y-3 = 1/7 (x-0)
 y = (1/7)x+3
the line tangent and the graph of : f passes through the point A(7, 4)
x= 7 y=4 so : f(7) =4

Final answer:

To find f(7) and f'(7) of y = f(x) given a point on the tangent line passing through another point.

Explanation:

Given:
Equation of tangent line to y = f(x) at (7, 4) passes through (0, 3).
To find:
f(7) and f'(7).

Find slope of tangent line using the two points: m = (3-4)/(0-7) = 1/7.Since slope of tangent = f'(7), f'(7) = 1/7.Using the point (7, 4) and slope 1/7, find the equation of the tangent line: y - 4 = (1/7)(x - 7).Substitute x = 7 in the equation to find f(7), which results in f(7) = 4.

Which line is parallel to the line y=12x+5 and passes through the point (-2, 1)?

Answers

The equation of the line parallel to y=12x+5 that passes through the point (-2, 1) is y=12x+25.

Since the two lines are parallel, they will have the same slope. The slope of the given line is the coefficient of x, which is 12. Therefore, the line we are looking for also has a slope of 12.

To find the equation of the new line, we'll use the point-slope form of a line equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.

Substituting the given point and the slope into this form we get:

y - 1 = 12(x + 2)

Expanding and simplifying this, we have:

y - 1 = 12x + 24

Finally, adding 1 to both sides to solve for y, we get:

y = 12x + 25, which is the equation of the line that is parallel to y=12x+5 and passes through the point (-2, 1).

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