Plug the values (–12, –7) and (12, 3) into the distance formula:

\(Distance = \sqrt{\left (x_{2} - x_{1} \right )^{2} + \left (y_{2} - y_{1} \right )^{2}}\) = \(\sqrt{\left (12 - \left (-12 \right ) \right )^{2} + \left (3 - \left (-7 \right ) \right )^{2}}\)

Now simplify to get the answer:

\(= \sqrt{\left (24 \right )^{2} + \left (10 \right )^{2}} = \sqrt{576 + 100} = \sqrt{676} = 26\)

The determinant of a matrix is a number, not a matrix, so rule out Choices (H), (J), and (K).

To determine which of the remaining answers is correct, use the determinant formula ad – bc:

(3 × 2) – (–6 × 1)

Simplify:

= 6 – (–6) = 6 + 6 = 12

The x-intercept of a line is the point where it crosses the x-axis — that is, where y = 0 — so the second coordinate must equal 0.

Therefore, you can rule out Choices (A) and (B).

To choose correctly among the three remaining answers, substitute 0 for y in the equation and solve for x:

y = 2x – 8

0 = 2x – 8

8 = 2x

4 = x

Thus, the x-intercept is (4, 0).

Let x be the original price of the guitar before tax.

Antoine paid $588.60 with a 9% percent increase on top of the original price.

Therefore, Antoine paid 109% of the original price, so you can create this equation:

1.09x = 588.60

Solve for x:

\(x = \frac{588.60}{1.09} = 540\)

To begin, note that ΔSUR is a right triangle with a leg the length of 9 and a hypotenuse the length of 15, so it’s a 9-12-15 version of a 3-4-5 triangle; therefore, SU = 12.

ΔTUS is a right triangle with a leg the length of 12 and a hypotenuse the length of 13, so it’s a 5-12-13 triangle; therefore, UT = 5.

So the base of ΔRST is 14, and its height is 12.

Plug these values into the formula for the area of a triangle to get your answer:

A = ¹⁄₂bh = ¹⁄₂(14)(12) = 84

Use the function you found for Question 29:

f(x) = 0.1x + 10

Plug in 100 for x and solve for x:

100 = 0.1x + 10

90 = 0.1x

900 = x

Danielle pays $0.10 per minute, so the function includes 0.1x.

However, she gets 200 minutes with her initial $30, so the input x needs to be changed to x– 200 to account for this.

Therefore, the function includes 0.1(x – 200).

Additionally, she’s charged $30, so the function is:

f(x) = 0.1(x – 200) + 30

Simplify by distributing and combining to get the final function:

f(x) = 0.1x – 20 + 30

f(x) = 0.1x + 10

The area of the park is 67,500 square feet.

If you let w equal the width of the park, the length is 3w.

Plug these numbers into the area formula for a rectangle and solve for w:

A = lw

67,500 = (3w)(w)

67,500 = 3w²

22,500 = w²

150 = w

If the width is 150 feet, the length is 150 × 3 = 450 feet.

Plug these numbers into the formula for the perimeter of a rectangle:

P = 2l + 2w = 2(450) + 2(150) = 900 + 300 = 1,200

The perimeter of the park is 1,200 feet.

Jane ran at 10 feet per second, so she ran for 120 seconds (because 1,200 ÷ 10 = 120).

To begin, notice that the first term of p³q⁴ + p⁴q⁵ contains (pq)³ multiplied by an extra q, and the second term contains (pq)⁴ multiplied by an extra q.

As a result, you can factor those values out of each respective term to simplify:

p³q⁴ + p⁴q⁵ = (pq)³q + (pq)⁴q

Now you can substitute 3 for pq and simplify:

= (3)³q + (3)⁴q

= 27q + 81q

= 108q

Remember that tanθ = ^O⁄_A, So if tanθ = ⁴⁄₃, the opposite and adjacent sides of the triangle are in a ratio of 4:3.

Thus, the triangle is a 3-4-5 triangle (you can verify this with the Pythagorean theorem), so you can make the following sketch:

Use this figure and the formula for the sine to answer the question:

sinθ = ^O⁄_H = ⁴⁄₅