Question

seiki is a city planner. she is planning to install a high - school regulation rectangular basketball court in a local park. she draws a blueprint that is the same shape as the court for the cement contractor. on the blueprint, 3 inches represents 20 feet on the court. the dimensions of the regulation basketball court are: - the length is 84 feet. - the width is 50 feet. enter the scale. \\(\frac{\text{blueprint}}{\text{court}} = \frac{3\text{ in.}}{20\text{ ft}} = 3\text{ in.} : 20\text{ ft}\\) enter the length, \\(b\\), of the blueprint. \\(\square\\) inches enter the width, \\(w\\), of the blueprint. \\(\square\\) inches i want to do this optional task to determine the length, \\(b\\). write a proportion using the variable \\(b\\). then solve for \\(b\\) using means and extremes.

Explanation:

Step1: Set up the proportion for length

We know the scale is 3 inches (blueprint) to 20 feet (court). Let \( b \) be the length on the blueprint (in inches) corresponding to 84 feet on the court. So the proportion is \(\frac{3}{20}=\frac{b}{84}\).

Step2: Solve for \( b \) using means and extremes

In a proportion \(\frac{a}{b}=\frac{c}{d}\), the product of the means (\(b\times c\)) equals the product of the extremes (\(a\times d\)). So for \(\frac{3}{20}=\frac{b}{84}\), we have \(20\times b = 3\times84\).
Calculate \(3\times84 = 252\), so \(20b = 252\). Then \(b=\frac{252}{20}=12.6\).

Step3: Set up the proportion for width

Let \( w \) be the width on the blueprint (in inches) corresponding to 50 feet on the court. The proportion is \(\frac{3}{20}=\frac{w}{50}\).

Step4: Solve for \( w \) using means and extremes

Using the means - extremes property, \(20\times w=3\times50\).
Calculate \(3\times50 = 150\), so \(20w = 150\). Then \(w=\frac{150}{20}=7.5\).

Answer:

The length \( b \) of the blueprint is \( 12.6 \) inches.
The width \( w \) of the blueprint is \( 7.5 \) inches.