Question

solve the following applications:

1. the perimeter of a rectangle is 86 feet. if the length is five feet less than twice the width, what are the dimensions of the rectangle.
2. in a triangle, the first angle is three times the measure of the third angle and the second angle is 50° more than the third angle. what are the measures of the three angles?

Explanation:

Problem 1:

Step1: Define variables

Let the width of the rectangle be \( w \) feet. Then the length \( l = 2w - 5 \) feet (since length is five feet less than twice the width).

Step2: Use perimeter formula

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). We know \( P = 86 \) feet. Substitute \( l = 2w - 5 \) into the formula:

$$ 86 = 2((2w - 5) + w) $$

Step3: Simplify and solve for \( w \)

First, simplify the expression inside the parentheses:

$$ 86 = 2(3w - 5) $$

Distribute the 2:

$$ 86 = 6w - 10 $$

Add 10 to both sides:

$$ 96 = 6w $$

Divide both sides by 6:

$$ w = 16 $$

Step4: Find the length

Substitute \( w = 16 \) into the length formula \( l = 2w - 5 \):

$$ l = 2(16) - 5 = 32 - 5 = 27 $$

Step1: Define variables

Let the measure of the third angle be \( x \) degrees. Then the first angle \( = 3x \) degrees and the second angle \( = x + 50 \) degrees.

Step2: Use triangle angle sum property

The sum of the interior angles of a triangle is \( 180^\circ \). So:

$$ 3x + (x + 50) + x = 180 $$

Step3: Simplify and solve for \( x \)

Combine like terms:

$$ 5x + 50 = 180 $$

Subtract 50 from both sides:

$$ 5x = 130 $$

Divide both sides by 5:

$$ x = 26 $$

Step4: Find the other angles

First angle: \( 3x = 3(26) = 78^\circ \)
Second angle: \( x + 50 = 26 + 50 = 76^\circ \)
Third angle: \( x = 26^\circ \)

Answer:

The width of the rectangle is 16 feet and the length is 27 feet.

Problem 2: