7th Grade Math

58 Expressions and Patterns

Numbers like √2 , √ 50 , and √ 78 are not perfect squares. These

numbers have square roots that are irrational numbers. A number

that cannot be expressed as the ratio of two integers is an irrational

number.

SCIENCE Physicist and mathematician Galileo performed an

experiment by dropping stones from the edge of the Leaning

Tower of Pisa. The formula t =

√h

_ 4 represents the time t in seconds

that it takes an object to fall from a height of h feet. If the tower is

180 feet tall, estimate the time it took the stones to hit the ground.

First estimate the value of √ 180.

169 < 180 < 196 169 and 196 are the closest perfect squares.

132 < 180 < 142 169 = 13^2 and 196 = 14^2

√ 13 ^2 < √  180 < √  142 Find the square root of each number.

13 < √  180 < 14 Simplify.

Since 180 is closer to 169 than 196, the best whole number estimate

for √ 180 is 13. Use this value to evaluate the expression.

t =

√ h

_ 4 ≈ _^134 or 3.25

So, stones dropped from the Leaning Tower of Pisa would take

about 3.25 seconds to hit the ground.

b. Estimate the time for an object to fall if dropped from the

observation deck of the Eiffel Tower at 311 feet.

Everyday Use

Irrational lacking usual or

normal clarity, as in

irrational thinking

Math Use

Irrational a number that

cannot be expressed as the

ratio of two integers

Example 1 Estimate each square root to the nearest whole number.

1. √  39 2. √  106 33 √  90 4. √  140


2. MEASUREMENT The diagram shows the floor plan of a

x

Area = x

105 ft^2

square kitchen. What is the approximate length of one

side of the kitchen floor? Round to the nearest whole

number.

Example 2 6. SPEED A vehicle’s speed can be estimated based on the

length of the skid marks made by the car tires using the

formula s = √ 24 m , where s represents the speed in miles per hour and m

represents the length of the skid in feet. A car leaves a mark that is 10 feet

long. Estimate the speed of the vehicle.

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