Question

1. the perimeter of a rectangle 28.8 centimeters. the length of the rectangle is twice as long as its width. find the length and width of the rectangle.
2. the area of a triangle is 338 square yards. the height of the triangle is four times its base. find the height and base of the triangle.

Explanation:

Problem 26

Step1: Define variables

Let the width of the rectangle be \( w \) (in cm). Then the length \( l = 2w \) (since length is twice the width).

Step2: Use perimeter formula

The perimeter of a rectangle is given by \( P = 2(l + w) \). We know \( P = 28.8 \) cm. Substitute \( l = 2w \) into the formula:
\[
28.8 = 2(2w + w)
\]

Step3: Simplify and solve for \( w \)

Simplify the right - hand side: \( 2(2w + w)=2(3w) = 6w \). So we have the equation \( 6w=28.8 \).
Divide both sides by 6: \( w=\frac{28.8}{6}=4.8 \) cm.

Step4: Find the length

Since \( l = 2w \), substitute \( w = 4.8 \) into this formula: \( l = 2\times4.8 = 9.6 \) cm.

Step1: Define variables

Let the base of the triangle be \( b \) (in yards). Then the height \( h = 4b \) (since height is four times the base).

Step2: Use area formula

The area of a triangle is given by \( A=\frac{1}{2}bh \). We know \( A = 338 \) square yards. Substitute \( h = 4b \) into the formula:
\[
338=\frac{1}{2}\times b\times(4b)
\]

Step3: Simplify and solve for \( b \)

Simplify the right - hand side: \( \frac{1}{2}\times b\times(4b)=2b^{2} \). So we have the equation \( 2b^{2}=338 \).
Divide both sides by 2: \( b^{2}=\frac{338}{2} = 169 \).
Take the square root of both sides: \( b=\sqrt{169}=13 \) yards (we take the positive root since length cannot be negative).

Step4: Find the height

Since \( h = 4b \), substitute \( b = 13 \) into this formula: \( h=4\times13 = 52 \) yards.

Answer:

The width of the rectangle is \( 4.8 \) cm and the length is \( 9.6 \) cm.

Problem 27