Radical Word Problems

Show your work for each problem.

1.
A ladder is 2.5 meters long. Its top reaches 2.4 meters up a wall. How far is the foot of the ladder from the wall?
1. a)3.47 m
2. b)0.1 m
3. c)0.7 m
4. d)0.49 m
Answer: 0.7 m
Using Pythagoras: $a^2 + 2.4^2 = 2.5^2$ , so $a^2 = 6.25 - 5.76 = 0.49$ and $a = \sqrt{0.49} = 0.7$ m. (0.49 forgets the square root, 0.1 subtracts the lengths, 3.47 adds the squares.)
2.
A square garden has an area of 144 square meters. What is the side length?
Answer: 12
Since Area $= s^2$ , we have $s = \sqrt{144} = 12$ meters.
3.
What formula would you use to find the diagonal of a rectangle with length 8 and width 6?
1. a) $d = \sqrt{8 \times 6}$
2. b) $d = 8 \times 6$
3. c) $d = \sqrt{8^2 + 6^2}$
4. d) $d = 8 + 6$
Answer: $d = \sqrt{8^2 + 6^2}$
The diagonal is the hypotenuse of a right triangle, so we use Pythagorean Theorem: $d = \sqrt{8^2 + 6^2}$ .
4.
A rectangular TV screen is 32 inches wide and 24 inches tall. Find the diagonal (screen size).
Answer: 40
1. What formula do we use? Pythagorean Theorem
2. Calculate $32^2$ 1024
3. Calculate $24^2$ 576
4. Add the squares: $1024 + 576$ 1600
5. Take the square root: $\sqrt{1600}$ 40
5.
Find the distance between points $(0, 0)$ and $(3, 4)$ .
Answer: 5
$d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
6.
A square has an area of 72 square cm. What is the side length in simplest radical form?
1. a) $9\sqrt{2}$ cm
2. b) $6\sqrt{2}$ cm
3. c) $\sqrt{72}$ cm
4. d) $2\sqrt{18}$ cm
Answer: $6\sqrt{2}$ cm
$s = \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$ cm. $2\sqrt{18}$ and $\sqrt{72}$ have the same value but are not fully simplified; $9\sqrt{2}$ squared is 162.
7.
Find the distance between points $A(2, 3)$ and $B(8, 11)$ .
Answer: 10
1. Find $x_2 - x_1$ (difference in x-coordinates) 6
2. Find $y_2 - y_1$ (difference in y-coordinates) 8
3. Calculate $(x_2 - x_1)^2$ 36
4. Calculate $(y_2 - y_1)^2$ 64
5. Add: $36 + 64$ 100
6. Take square root: $\sqrt{100}$ 10
8.
A kite string is 25 meters long. The kite is directly above a point 7 meters from the person holding the string. How high is the kite?
Answer: 24
Using $h^2 + 7^2 = 25^2$ : $h^2 + 49 = 625$ , so $h^2 = 576$ and $h = 24$ meters.
9.
A town square has the shape of a square with 90-meter sides. A straight path crosses it from one corner to the opposite corner. How long is the path? Give the answer in simplest radical form.
Answer: $90\sqrt{2}$
1. Which part of the square is the path? diagonal
2. Calculate $90^2$ 8100
3. For the diagonal: $d^2 = 90^2 + 90^2 = 2 \times 8100$ . What is $2 \times 8100$ ? 16200
4. $\sqrt{16200} = \sqrt{8100 \times 2} = \square\sqrt{2}$ . What number goes in the box? 90
10.
The sloping side of a barn roof rises 5 meters over a horizontal distance of 12 meters. How long is the sloping roof edge?
1. a)15 m
2. b)13 m
3. c)17 m
4. d) $\sqrt{119}$ m
Answer: 13 m
$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ m. This is the 5-12-13 Pythagorean triple! (17 adds the lengths, $\sqrt{119}$ subtracts the squares.)
11.
A rectangular pool is 15 meters by 8 meters. A swimmer swims diagonally across. What is the exact distance?
Answer: 17
1. Calculate $15^2$ 225
2. Calculate $8^2$ 64
3. Add: $225 + 64$ 289
4. What is $\sqrt{289}$ ? 17
12.
Find the distance between $(1, 5)$ and $(4, 11)$ . In simplest radical form it is $a\sqrt{5}$ . What is $a$ ?
Answer: 3
$d = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$ , so $a = 3$ .
13.
A 20-meter cable runs from the top of a mast to a stake in the ground. The stake is 16 meters from the foot of the mast. How tall is the mast?
Answer: 12
1. The cable is 20 meters long. Which part of the right triangle is it? hypotenuse
2. Calculate $20^2$ 400
3. Calculate $16^2$ 256
4. For height: $h^2 = 20^2 - 16^2$ . Calculate $400 - 256$ 144
5. What is $\sqrt{144}$ ? 12
14.
Find the distance between $(-2, 3)$ and $(1, -1)$ .
1. a) $\sqrt{7}$
2. b) $\sqrt{5}$
3. c)7
4. d)5
Answer: 5
$d = \sqrt{(1-(-2))^2 + (-1-3)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
15.
A helicopter hovers at a height $h$ . A straight rescue line, 130 meters long, is stretched from the helicopter to a boat on the water. The boat is 50 meters from the point directly below the helicopter. How high is the helicopter?
Answer: 120
1. Which part of the right triangle is the line (130 m)? hypotenuse
2. Calculate $130^2$ 16900
3. Calculate $50^2$ 2500
4. $h^2 = 130^2 - 50^2$ . Calculate $16900 - 2500$ 14400
5. What is $\sqrt{14400}$ ? 120
16.
A square has a diagonal of $10\sqrt{2}$ cm. What is the side length?
Answer: 10
Since $d = s\sqrt{2}$ , we have $s = \frac{d}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10$ cm.