Word Problems
Overview
Apply your understanding of square roots, irrational numbers, and the real number system to solve real-world problems. These contexts help you see where irrational numbers appear in everyday life.
Practice Problems
Question 1: A square garden has an area of 200 square feet. What is the length of each side? Give an exact answer and an approximation.
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Answer: Exact: sqrt(200) = 10sqrt(2) feet; Approximate: 14.14 feet
Side = sqrt(Area) = sqrt(200) = sqrt(100 x 2) = 10sqrt(2) feet, which is about 14.14 feet.
Question 2: A cube-shaped box has a volume of 64 cubic inches. What is the length of each edge?
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Answer: 4 inches
Edge = cube root of volume = cube root of 64 = 4 inches (since 4 x 4 x 4 = 64).
Question 3: The diagonal of a square TV screen is sqrt(2) times the side length. If a TV has 40-inch sides, what is the diagonal length?
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Answer: 40sqrt(2) inches, approximately 56.57 inches
Diagonal = 40 x sqrt(2) = 40sqrt(2) inches, which is about 56.57 inches.
Question 4: A physicist calculates the time for an object to fall as t = sqrt(2h/g). If h = 45 meters and g = 10 m/s^2, find t.
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Answer: 3 seconds
t = sqrt(2 x 45 / 10) = sqrt(90/10) = sqrt(9) = 3 seconds.
Question 5: A circular pizza has area 50pi square inches. What is its radius?
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Answer: sqrt(50) = 5sqrt(2) inches, approximately 7.07 inches
A = pi*r^2, so 50pi = pi*r^2. Then r^2 = 50, so r = sqrt(50) = 5sqrt(2) inches.
Question 6: Two friends argue about whether sqrt(2) x sqrt(8) is rational or irrational. Who's right and why?
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Answer: It's rational (equals 4)
sqrt(2) x sqrt(8) = sqrt(16) = 4, which is rational. The product of two irrationals can be rational!
Question 7: A pendulum's period is T = 2pi*sqrt(L/g). If L = 1 meter and g = 10 m/s^2, is T rational or irrational?
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Answer: Irrational
T = 2pi*sqrt(1/10) = 2pi*sqrt(0.1). Since pi is irrational, any multiple of pi is irrational.
Question 8: A rope is exactly sqrt(50) meters long. A student says it's "about 7 meters." Is this a reasonable estimate?
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Answer: Yes, it's reasonable
sqrt(50) is between sqrt(49) = 7 and sqrt(64) = 8. Actually sqrt(50) is about 7.07, so "about 7" is reasonable.
Question 9: A cube has surface area 96 square cm. What is its volume?
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Answer: 64 cubic cm
Surface area = 6s^2 = 96, so s^2 = 16, s = 4 cm. Volume = s^3 = 4^3 = 64 cubic cm.
Question 10: The distance formula gives d = sqrt((x2-x1)^2 + (y2-y1)^2). Between (0,0) and (1,1), is the distance rational or irrational?
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Answer: Irrational
d = sqrt((1-0)^2 + (1-0)^2) = sqrt(1+1) = sqrt(2), which is irrational.