The conjugate of 2 + 7i is 2 – 7i, so you’re looking for the value of (2 + 7i)(2 – 7i).

Begin by FOILing to remove the parentheses:

(2 + 7i)(2 – 7i) = 4 – 14i + 14i – 49i² = 4 – 49i²

Now, because i = √(-1), you can substitute –1 for i²:

4 – 49(–1) = 4 + 49 = 53

The water level decreases and then rises again, which rules out Choices (A) and (C).

The pool is full to capacity at the beginning of the shift, so it can’t have more water at the end of the shift; therefore, Choice (D) is also wrong.

Finally, the pool drains in 2 hours but takes 7 hours to fill, so the downward slope at the beginning of the shift is greater than the upward slope at the end; as a result, you can rule out Choice (B), leaving Choice (E) as the correct answer.

The function f(x) is symmetrical, so if you reflect it horizontally across the y-axis and then shift it 6 units to right, it returns to where it started.

To reflect it horizontally, change x to –x:

f(–x)

Now, to move this function 6 units to the right, change x in this new function to x – 6 and simplify:

f(–(x – 6)) = f(–x + 6) = f(6 – x)

The octagon is regular, so the shaded region is a right triangle.

It has a base of 1, so you need to know its height to find its area.

Draw a few lines as follows to help find the height of the triangle:

If you let x equal the unknown length, the height of the shaded region is 2x + 1.

To find the value of x,notice that the length x is the leg of a 45-45-90 triangle with a hypotenuse of 1.

The ratio of a leg of this triangle to its hypotenuse is 1 : √2

Thus:

\(\frac{1}{\sqrt{2}} = \frac{x}{1}\)

Therefore:

\(x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)

So

2x + 1 = √2 + 1

Plug this value as the height into the formula for the area of a triangle, with a base of 1:

\(A = \frac{1}{2}bh = \frac{1}{2}\left ( 1 \right )\left ( \sqrt{2} + 1 \right ) = \frac{\sqrt{2} + 1}{2}\)

If Victoria doubles her registration to 40 and the others keep the same numbers, the sum of registrations will be:

12 + 16 + 19 + 13 + 40 = 100

Thus, Victoria’s registration will be:

⁴⁰⁄₁₀₀ = 40%

Zach registered 12 clients, Yolanda registered 16, Xavier registered 19, Wanda registered 13, and Victoria registered 20.

Add these together to find the total number of clients registered:

12 + 16 + 19 + 13 + 20 = 80

So Yolanda registered:

¹⁶⁄₈₀ = ¹⁄₅ = 20%

Let p, q, and r equal the amounts that Paulette, Quentin, and Rosie gave, respectively.

Then you can set up the following three equations:

p = q + r ; 3q = p + 40 ; 2(r – 20) = p

Simplify the third equation

p = q + r ; 3q = p + 40 ; 2r – 40 = p

Substitute q + r for p into the second and third equations:

3q = q + r + 40 ; 2r – 40 = q + r

Simplify both equations:

2q = r + 40; r – 40 = q

Now substitute r – 40 for q into the equation 2q = r + 40 and solve for r:

2(r – 40) = r + 40

2r – 80 = r + 40

r – 80 = 40

r = 120

Substitute 120 for r in the equation 2r – 40 = p and solve for p:

2(120) – 40 = p

240 – 40 = p

200 = p

The height of the tank is hand its diameter is 3h, so its radius is 1.5h.

The volume of the tank is approximately 231.5 cubic meters.

Use 3.14 as an approximation of π and plug these values into the formula for a cylinder:

V = πr²h

231.5 = (3.14)(1.5h)²h

Simplify and solve for h:

231.5 ≈ (3.14)(2.25h²)h

231.5 ≈ 7.065h³

32.767 ≈ h³

3.2 ≈ h

Thus, the height of the tank is closest to 3 meters.

First, use the reciprocal identity \(sec x = \frac{1}{cos x}\) to substitute for sec x:

\(sin x sec x = \frac{sin x}{cos x}\)

Now recall the following identity:

\(\frac{sin x}{cos x} = tan x\)

The formula for a circle of radius r is (x – h)² + (y – k)² = r².

So in the equation (x + 3)² + (y – 4)² = 49:

r² = 49

r = 7

You now plug this value into the formula for the area of a circle to get your answer:

A = πr² = π(7)² = 49 π