Question

5 the formula for the circumference of a circle is ( c = 2pi r ). which equation below would solve the equation for ( r )? a ( r = 2pi c ) b ( r=\frac{1}{2}pi c ) c ( r=\frac{c}{2pi} ) d ( r=\frac{2pi}{c} ) 6 a triangle has a base of 40 inches and a height of ( (2x + 8) ) inches. a rectangle has a length of ( (5x - 4) ) inches and a width of 20 inches. the area in square inches of the triangle is equal to the area in square inches of the rectangle. what is the value of ( x )? a 4 b 20 c ( \frac{4}{3} ) d 12

Explanation:

5. Solve for \( r \) in \( C = 2\pi r \)

Step1: Divide both sides by \( 2\pi \)

Given \( C = 2\pi r \), to isolate \( r \), we use the property of equality. Divide each side of the equation by \( 2\pi \).

$$ \frac{C}{2\pi}=\frac{2\pi r}{2\pi} $$

Step2: Simplify the right - hand side

On the right - hand side, \(\frac{2\pi r}{2\pi}\) simplifies to \( r \) (since \( 2\pi\div2\pi = 1 \)). So \( r=\frac{C}{2\pi}\)

6. Set up the area equations

Step1: Recall the area formulas

The area of a triangle \(A_{t}=\frac{1}{2}\times\text{base}\times\text{height}\), and the area of a rectangle \(A_{r}=\text{length}\times\text{width}\)
For the triangle, with base \(b = 40\) inches and height \(h=(2x + 8)\) inches, \(A_{t}=\frac{1}{2}\times40\times(2x + 8)\)

$$ A_{t}=20\times(2x + 8)=40x+160 $$

For the rectangle, with length \(l=(5x - 4)\) inches and width \(w = 20\) inches, \(A_{r}=20\times(5x - 4)=100x-80\)

Step2: Set the areas equal

Since \(A_{t}=A_{r}\), we have the equation \(40x + 160=100x-80\)

Step3: Solve for \(x\)

Subtract \(40x\) from both sides:

$$ 40x+160-40x=100x - 80-40x $$

$$ 160 = 60x-80 $$

Add \(80\) to both sides:

$$ 160+80=60x-80 + 80 $$

$$ 240=60x $$

Divide both sides by \(60\):

$$ x=\frac{240}{60}=4 $$

Answer:

1. C. \(r=\frac{C}{2\pi}\)
2. A. \(4\)