Question

question
the rectangle below is dilated by a scale factor of \\( \frac { 1 } { 5 } \\). find the perimeter and area of the rectangle below, as well as the perimeter and area of the dilated rectangle. express your answers as whole numbers, decimals, or fractions in simplest terms (no mixed numbers). figures are not necessarily drawn to scale.
answer
perimeter of given rectangle units
area of given rectangle units²
perimeter of dilated rectangle units
area of dilated rectangle units²

Explanation:

Step1: Calculate the perimeter of the given rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\). Here, \(l = 70\) and \(w = 35\).

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Step2: Calculate the area of the given rectangle

The formula for the area of a rectangle is \(A=l\times w\). Here, \(l = 70\) and \(w = 35\).

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Step3: Calculate the dimensions of the dilated rectangle

When a figure is dilated by a scale factor \(k=\frac{1}{5}\), the new length \(l_{new}=l\times k\) and new width \(w_{new}=w\times k\). So, \(l_{new}=70\times\frac{1}{5} = 14\) and \(w_{new}=35\times\frac{1}{5}=7\)

Step4: Calculate the perimeter of the dilated rectangle

Using the perimeter formula \(P = 2(l + w)\) with \(l = 14\) and \(w = 7\)

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Step5: Calculate the area of the dilated rectangle

Using the area formula \(A=l\times w\) with \(l = 14\) and \(w = 7\)

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Answer:

Perimeter of given rectangle: \(210\) units
Area of given rectangle: \(2450\) units²
Perimeter of dilated rectangle: \(42\) units
Area of dilated rectangle: \(98\) units²