A circular wool quilt of 1.2 m diameter costs 200 . What should the price of a 1.6-m -diameter

step 1 We have cost is proportional to r square So we have cost over cost prime equals r over r prime S

A circular wool quilt of 1.2 m diameter costs 200 . What...

A circular wool quilt of 1.2 $\mathrm{m}$ diameter costs $\$ 200 .$ What should the price of a $1.6-\mathrm{m}$ -diameter quilt be if it is to have the same cost per unit area? $\begin{array}{lllll}{\text { (a) } \$ 110} & {\text { (b) } \$ 150} & {\text { (c) } \$ 270} & {\text { (d) } \$ 360}\end{array}$

A circular wool quilt of 1.2 $\mathrm{m}$ diameter costs $\$ 200 .$ What
should the price of a $1.6-\mathrm{m}$ -diameter quilt be if it is to have
the same cost per unit area?
$\begin{array}{lllll}{\text { (a) } \$ 110} & {\text { (b) } \$ 150} & {\text { (c) } \$ 270} & {\text { (d) } \$ 360}\end{array}$

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Key Concepts

Unit Rate Calculation

This concept involves determining the cost per unit of area by dividing the total cost by the total area. Once the unit rate is known, it can be applied to other shapes or sizes to find the corresponding total cost, assuming the rate remains constant.

Direct Proportionality

Direct proportionality is the relationship where two quantities increase or decrease at the same rate. In geometric contexts, such as this problem, a constant cost per unit area means the total cost is directly proportional to the area of the shape.

Area of a Circle

This concept involves calculating the area of a circle using the formula A = ?r². Understanding this is essential for determining the space enclosed within a circular boundary, which is a foundational element of geometric analysis.

Scaling of Area with Diameter

This key concept explains that when the diameter or radius of a circle changes, the area does not change linearly but rather with the square of the change factor. It is critical for understanding how a change in one dimension—such as diameter—affects the overall area.

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