Radical Word Problems
1
The radius in inches of a balloon can be expressed as
3
𝑟= √
3𝑉
4𝜋
where r is the radius and V is the volume of the balloon in cubic inches. If air is pumped to inflate the balloon
from 500 cubic inches to 800 cubic inches, by how many inches has the radius increased?
What was the radius of the balloon originally?
What was the radius after inflating the balloon to 800 cubic inches?
How can you use the two radii to find the amount of increase?
2
The voltage V of an audio system’s speaker can be represented by 𝑉 = 4√𝑃, where P is the power of the
speaker. An engineer wants to design a speaker with 400 watts of power. What will the voltage be?
3
Boat builders share an old rule of thumb for sail boats. The maximum speed K in knots is about 1.35 times the
square root of the length L in feet of the boats waterline.
a. A customer is planning to order a sailboat with a maximum speed of 12 knots. How long should the
water line be?
b. How much longer would the waterline have to be to achieve a maximum speed of 15 knots?
1
Radical Word Problems
4
2𝑠
The formula 𝑡 = √ 𝑎 shows the time t that any vehicle takes to travel a distance s at a constant acceleration a,
starting from rest. What is the difference in time between a car accelerating at 16 m/s2 and one accelerating
25m/s2 for a distance of 200m?
What is the time that a car accelerating at 16 m/s2 takes to travel 200m?
What is the time that a car accelerating at 25 m/s2 takes to travel 200m?
5
The circular velocity V in miles per hour of a satellite orbiting the Earth is given by the formula
1.24×1012
𝑟
𝑣=√
Where r is the distance in miles from the satellite to the center of the Earth. How much greater is the velocity
of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi.
(The radius of the Earth is 3950 mi.)
6
The base of a triangle is √18 cm and its height is √8 cm. Find its area.
7
The radius of a circle can be expressed as 𝑟 = √ inches where r is the radius and A is the area of the circle. If
𝐴
𝜋
the area of a circle is 169𝜋 in2, what is its radius?
2
Radical Word Problems
8
3
𝑉
For Meteor Crater in Arizona, the formula 𝑑 = 2√0.3 relates the diameter d of the rim (in meters) to the
volume V (in cubic meters). Given the diameter of Meteor Crater is 1.2 kilometers, calculate the volume.
9
A spherical water tank holds 9000 ft3 of water. Using the formula = √ 𝜋 , find the diameter of the tank.
10
The formula 𝑑 = √
3
4𝑄
𝜋𝑉
6𝑉
models the diameter of a pipe where Q is the maximum flow of a water pipe , and v is the
velocity of the water. What is the diameter of a pipe that allows a maximum flow of 30 ft3/min of water
flowing at a velocity of 400 ft/min?
11
The velocity v of an object dropped from a tall building is given by the formula 𝑣 = √64𝑑 where d is the
distance the object has dropped. Solve the formula for d.
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Radical Word Problems
12
𝐿
The formula 𝑠 = 2𝜋√32 represents the swing of a pendulum. S is the time in seconds to swing back and forth,
and L is the length of the pendulum in feet.
a.) How long does it take for a 3 foot pendulum to swing back and forth? (Round to three decimal places.)
b.) Solve the formula for L
c.) Find the length of a pendulum that makes one swing in 2.5 seconds. (Round to three decimal places.)
13
14
The speed that a tsunami (tidal wave) can travel is modeled by the equation 𝑆 = 356√𝑑 where S is the speed in
kilometers per hour and d is the average depth of the water in kilometers.
a.) What is the speed of the tsunami when the average water depth is 0.512 kilometers? (round to the
nearest tenth)
b.) Solve the equation for d.
c.) A tsunami is found to be traveling at 120 kilometers per hour. What is the average depth of the water?
(round to three decimal places)
Isaac Newton established the formula 𝑉 = √
2𝐺𝑀
𝑅
to calculate the escape velocity from a planet or star where V
is the escape velocity, G is the gravitational Constant, M is the mass of the planet or star and R is the radius of
the planet or star. Solve the equation for radius of the planet or star.
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Radical Word Problems
1225√𝑎
15
The distance a person can see on a clear day is estimated by the formula 𝑣 = 1000 where v=visibility in miles
and a = altitude. If Jerome can see for 150 miles, about how far above ground is he?
16
The average amount of apples consumed by Americans (in pounds per person) between 1980 and 2000 can be
modeled by the equation 𝑦 = √22𝑥 + 275 where x is the number of years since 1980. In what year were
about 20 pounds of apples consumed per person?
17
The Yellow Jackets have won a total of (23 + √−36) games, and the Bulldogs have won a total of (11 − √−9).
How many more games have the Yellow Jackets won?
18
A circuit has a current of (14 + 7𝑖) amps and another circuit has a current of (3 − 23𝑖) amps. What is the sum
of the currents of the two circuits?
19
A soccer field has a length of (√−7 + √15) yards and a width of (√−7) yards. If the area of the rectangle is
equal to length times width, what is the area of the soccer field?
20
The area of a big screen television (8 − 3𝑖) inches2. The length of the big screen is (4 + 7𝑖) inches. What is the
width of the television screen?
21
Jimmy is installing a tile floor in a room in his home. He needs to know how many tiles it will take to cover the
floor. If the length of the floor is (9 + √−18) inches and the width of the floor is √−40 inches, what is the
area of the floor?
22
Lisa wants to buy two new dresses from themall. If one dress costs (59 − 6𝑖) dollars and the other costs (34 +
17𝑖) dollars, how much money will she need to bring to the mall?
5
Radical Word Problems
6