Exercise 11 Page 483

For any two positive numbers

a

and

b,

the geometric mean is the positive number

x

such that the following equation holds true.

\frac{a}{x} = \frac{x}{b}

We are asked to find the geometric mean of the given pair of numbers.

10 and 15

Let's substitute them into the equation and use the Cross Product Property to find the mean

x .

\frac{1 0}{x} = \frac{x}{1 5}

Cross multiply

10 \cdot 15 = x \cdot x

▼

Solve for

x

Rearrange equation

x \cdot x = 10 \cdot 15

$a \cdot a = a^{2}$

x^{2} = 10 \cdot 15

x^{2} = 150

$LHS = RHS$

∣ x ∣ = 150

$x \geq 0$

x = 150

Split into factors

x = 25 \cdot 6

$a \cdot b = a \cdot b$

x = 25 \cdot 6

Calculate root

x = 56