MATHEMATICS HIGH SCHOOL

. Let A = (−2, 4) and B = (7, 6). Find the point P on the line y = 2 that makes the total distance AP + BP as small as possible.

Answers

Answer 1

Answer:

Answer:

P(1,2)

Step-by-step explanation:

There are 2 points.

A(-2,4) and B(7,6)

the point P on the y=2 can also represented as P(x,2)

We can use the distance formula to find the distances AP and BP

$\text{dist} = √((x_1 - x_2)^2 + (y_1 - y_2)^2)$

for AP: A(-2,4) and P(x,2)

$AP = √((-2 - x)^2 + (4 - 2)^2)$

$AP = √((-2 - x)^2 + 4)$

$AP = √((-1)^2(2 + x)^2 + 4)$

$AP = √((2 + x)^2 + 4)$

for BP: B(7,6) and P(x,2)

$BP = √((7 - x)^2 + (6 - 2)^2)$

$BP = √((7 - x)^2 + 16)$

the total distance AP + BP will be

$√((2 + x)^2 + 4)+√((7 - x)^2 + 16)$ (plot is given below)

Our task is to find the value of x such that the above expression is small as possible. (we can find this either through plotting or differentiating)

If you plot the above equation, the minimum point of the curve will be clearly visible, and it will be at x = 1. Hence, the point P(1,2) is such that the total distance AP + BP is as small as possible.

Answer 2

Answer:

Final answer:

The point P that makes the total distance AP + BP smallest on the line y=2 is given by the x-coordinate of the midpoint of A and B because the shortest distance is in a straight line. Therefore, the point P is (2.5, 2).

Explanation:

To find the point P on the line y = 2 that makes the total distance AP + BP the smallest, you need to recall that the shortest distance between two points is a straight line. So, ideally, we want to find a point P (x,2) that is on the same vertical line (or x-coordinate) that intersects the line AB at the midpoint.

Step 1: Find the midpoint of A and B. The midpoint M is obtained by averaging the x and y coordinates of A and B: M = ((-2+7)/2 , (4+6)/2) = (2.5, 5).

Step 2: Since line y = 2 is horizontal, the x-coordinate of our point P will stay the same with the midpoint x-coordinate. Therefore, P has coordinates (2.5, 2).

So, the point on the line y = 2 that makes the total distance AP + BP as small as possible is P (2.5, 2).

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Audrey is buying a new car for $32,998.00. She plans to make a down payment of $4,200.00. If she's to make monthly payments of $525 for the next five years, what APR has she paid?A. 37%
B. .37%
C. 3%
D. 3.7%

Answers

This question is an annuity problem with cost of the car = $32,998, the present value of the annuity (PV) is given by the difference between the cost of the car and the down payment = $32,998 - $4,200 = $28,798. The monthly payments (P) = $525 and the number of number of years (n) = 5 years and the number of payments in a year (t) is 12 payments (i.e. monthly) The formula for the present value of an annuity is given by PV = (1 - (1 + r/t)^-nt) / (r/t) 28798 = 525(1 - (1 + r/12)^-(5 x 12)) / (r/12) 28798r / 12 = 525(1 - (1 + r/12)^-60) 28798r / (12 x 525) = 1 - (1 + r/12)^-60 2057r / 450 = 1 - (1 + r/12)^-60 Substituting option A (r = 37% = 0.37) 2057r / 450 = 2057(0.37) / 450 = 761.09 / 450 = 1.691 1 - (1 + r/12)^60 = 1 - (1 + 0.37/12)^-60 = 1 - 0.1617 = 0.8383 Therefore, r is not 37% Substituting option D (r = 3.7% = 0.037) 2057r / 450 = 2057(0.037) / 450 = 76.109 / 450 = 0.1691 1 - (1 + r/12)^60 = 1 - (1 + 0.037/12)^-60 = 1 - 0.8313 = 0.1687 Therefore, r is approximately 3.7%

To solve this we are going to use the loan payment formula: $P= ( (r)/(n)(PV))/(1-(1+ (r)/(n))^(-nt) )$

where

$P$ is the amount of the regular payment

$PV$ is the present debt

$r$ is APR in decimal form

$n$ is the number of payments per year

$t$ is the time in years

We know from our problem that Audrey is making a down payment of $4,200.00; since the cost of the car is $32,998.00, the present deb will be the cost of the car minus the down payment, so $PV=32998-4200=28798$ . We also know that she is going to make monthly payments of $525 for the next five years, so $n=12$ and $t=12$ . Let's replace the values in our formula:

$P= ( (r)/(n)(PV))/(1-(1+ (r)/(n))^(-nt) )$

$525= ( (r)/(12)(28798))/(1-(1+ (r)/(12))^(-(12)(5)) )$

We have two ways of finding the APR: we can solve for $r$ in our equation, which is extremely difficult, or we can evaluate the given APRs and check for which one both sides of the equation are almost the same. Since the second is way easier, we are going to use it.

A. 37%

The APR should be in decimal form, so we need to convert it first; to do it we are going to divide the APR by 100%

$r=(37)/(100) =0.37$

Let's replace the ARP in decimal form in our equation

$525= ( (0.37)/(12)(28798))/(1-(1+ (0.37)/(12))^(-(12)(5)) )$

$525=1059.20$

$529\neq 1059.20$

Since 529 is not equal to 1059.20, 37% is not the APR of the loan.

B. .37%

- Convert the APR to decimal form

$r=(0.37)/(100) =0.0037$

- Replace the APR

$525= ( (0.0037)/(12)(28798))/(1-(1+ (0.0037)/(12))^(-(12)(5)) )$

$525=484.49$

Since 525 is not equal to 484.59, .37% is not the APR of the loan.

C. 3%

- Convert the APR to decimal form

$r=(3)/(100) =0.03$

- Replace the APR

$525= ( (0.03)/(12)(28798))/(1-(1+ (0.03)/(12))^(-(12)(5)) )$

$525=517.46$

Since 525 is not equal to 517.46, 3% is not the APR of the loan.

D. 3.7%

- Convert the APR to decimal form

$r=(3.7)/(100) =0.037$

- Replace the APR

$525= ( (0.037)/(12)(28798))/(1-(1+ (0.037)/(12))^(-(12)(5)) )$

$525=526.47$

Since 525 is almost equal to 526.47, 3.7% is the APR of the loan.

We can conclude that the correct answer is D. 3.7%

When factored completely, the expression p^4 - 81 is equivalent to(1) (p^2 + 9)(p^2 - 9)
(2) (p^2 - 9)(p^2 - 9)
(3) (p^2 + 9)(p + 3)(p - 3)
(4) (p + 3)(p - 3)(p + 3)(p - 3)

Answers

p⁴ - 81
p⁴ + 9p² - 9p² - 81
p²(p²) + p²(9) - 9(p²) - 9(9)
p²(p² + 9) - 9(p² + 9)
(p² - 9)(p² + 9)
(p² + 3p - 3p - 9)(p² + 9)
(p(p) + p(3) - 3(p) - 3(3))(p² + 9)
(p(p + 3) - 3(p + 3))(p² + 9)
(p - 3)(p + 3)(p² + 9)

The answer is (3).

Final answer:

The expression p^4 - 81, when factored completely, is equivalent to (p^2 + 9)(p^2 - 9) using the difference of squares formula.

Explanation:

When the expression p^4 - 81 is factored completely, it becomes equivalent to (p^2 + 9)(p^2 - 9). This can be calculated using the rule of difference of squares which states that a^2 - b^2 = (a - b)(a + b). Here, p^4 can be considered as a square of p^2 and 81 as a square of 9. Hence p^4 - 81 = (p^2)^2 - 9^2. Applying difference of squares formula we get (p^2 - 9)(p^2 + 9).

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Solve x+2=1/2x-1 Please hurry!

Answers

Answer:

X = - 6

Step-by-step explanation:

x + 2 = 1/2x - 1

$2 + 1 = (1)/(2)x - x$

$3 = - (1)/(2)x$

$x = ( - 3)/( (1)/(2) ) \n x = ( - 6)/(1) \n x = - 6$

Hopethishelps:)

If possible, please show solving steps

Answers

let 1/u=x and 1/v=y

so,

-3x-8y=20

-5x+y=19

then solve for it

When you find A and B, back-substitute for A and B and continue from there.

The average high temperature in Anchorage, Alaska, in January is 21°F with a standard deviation of 10°. The average high temperature in Honolulu in January is 80°F with a standard deviation of 8°. In which location would it be more unusual to have a day in January with a high of 57°F?

Answers

Answer:

In Honolulu it be more unusual to have a day in January with a high of 57°F

Step-by-step explanation:

Given,

The average high temperature in Anchorage, Alaska,

$\mu_1$ = 21° F,

Having standard deviation,

$\sigma_1$ = 10°,

Thus, the z-score for the high temperature of 57° F in alaska,

$z_1=(57-\mu_1)/(\sigma_1)$

$=(57-21)/(10)$

$=3.6$

By the z-score table the probability of the high temperature of 57°F in Alaska = 0.99984 = 99.984 %

Now, the average high temperature in Honolulu is,

$\mu_2=80^(\circ)$

Having standard deviation,

$\sigma_2=8^(\circ)$

Thus, the z-score for the high temperature of 57° F in Honolulu,

$z_2=(57-\mu_2)/(\sigma_2)$

$=(57-80)/(8)$

$=-2.875$

By the z-score table the probability of the high temperature of 57° in Honolulu = 0.00205 = 0.205 %

Since, a probability is unusual if it is less than 5 %,

Hence, In Honolulu it be more unusual to have a day in January with a high of 57°F.

It would be more odd in Honolulu.

Brent is making a playlist for an upcoming trip. The average length of a song is 3.53 minutes If the flight is about 120 minutes, how many songs should be put on his playlist

Answers

Answer:

34 songs

Step-by-step explanation:

120min/3.53 = 33.99

average length of a song = 3.53 minutes

time duration of flight = 120 minutes

no. of songs that should be put on his playlist

= 120/3.53

= 33.99....

= 34 ( approximately )

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