Question

solve each radical equation for the variable.

1. $sqrt{4w + 9} = sqrt{w + 21}$
2. $sqrt{3w - 5} = sqrt{w + 11}$
3. $sqrt{x - 8} - 6 = -2$
4. $sqrt{2x + 5} - 19 = -3$
5. on a roller coaster ride your speed depends on the height of the hill you have just come down and the radius of the loop in feet. the equation $v = 8sqrt{h - 2r}$ gives the velocity $v$ in feet per second of a car at the top of the loop. suppose the loop has a radius of 16 feet. you want the car to have a velocity of 64 ft/sec at the top of the loop. how high should the hill be?

Explanation:

Step1: Solve equation 22: Square both sides

$$(\sqrt{4w+9})^2 = (\sqrt{w+21})^2$$
$$4w + 9 = w + 21$$

Step2: Isolate variable w

$$4w - w = 21 - 9$$
$$3w = 12$$
$$w = \frac{12}{3} = 4$$

Step3: Solve equation 23: Square both sides

$$(\sqrt{3w-5})^2 = (\sqrt{w+11})^2$$
$$3w - 5 = w + 11$$

Step4: Isolate variable w

$$3w - w = 11 + 5$$
$$2w = 16$$
$$w = \frac{16}{2} = 8$$

Step5: Solve equation 24: Isolate the radical

$$\sqrt{x-8} = -2 + 6$$
$$\sqrt{x-8} = 4$$

Step6: Square both sides, solve x

$$(\sqrt{x-8})^2 = 4^2$$
$$x - 8 = 16$$
$$x = 16 + 8 = 24$$

Step7: Solve equation 25: Isolate the radical

$$\sqrt{2x+5} = -3 + 19$$
$$\sqrt{2x+5} = 16$$

Step8: Square both sides, solve x

$$(\sqrt{2x+5})^2 = 16^2$$
$$2x + 5 = 256$$
$$2x = 256 - 5 = 251$$
$$x = \frac{251}{2} = 125.5$$

Step9: Solve problem 26: Substitute given values

Given $v=64$, $r=16$, substitute into $v = 8\sqrt{h-2r}$:
$$64 = 8\sqrt{h - 2(16)}$$
$$64 = 8\sqrt{h - 32}$$

Step10: Isolate the radical

$$\sqrt{h - 32} = \frac{64}{8} = 8$$

Step11: Square both sides, solve h

$$(\sqrt{h - 32})^2 = 8^2$$
$$h - 32 = 64$$
$$h = 64 + 32 = 96$$

Answer:

1. $w=4$
2. $w=8$
3. $x=24$
4. $x=125.5$
5. The hill should be 96 feet high.