Answer
a. $450$ houses
b. $50$ cm (diameter needed)
Work Step by Step
Solving Variation Problems (see p. 424)
1. $\ \ $Write an equation that models the given English statement.
2. $\ \ $Substitute the given pair of values into the equation in step 1 and find the value of k, the constant of variation.
3. $\ \ $Substitute the value of k into the equation in step 1.
4. $\ \ $Use the equation from step 3 to answer the problem's question.
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Let h be the number of houses, d the pipe diameter (cm).
1.
$h$ varies directly as $d^{2},\ \qquad h=kd^{2}$
2.
$50=k(10^{2}) $
$50=k(100) \qquad /\div 100$
$k=0.5$
3.
$h=0.5d^{2}$
4$a$.
$d=3, h=?$
$h=0.5(30^{2})= 450$ houses
4b.
h=1250, $d=?$
$1250=0.5d^{2}\qquad/\times 2$
$2500=d^{2}$
$d=50$ cm (diameter needed)