Question

find the exact area.
the area is

Explanation:

Step1: Recall the formula for the area of a triangle

The area \( A \) of a triangle is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is \( 4\sqrt{20} \) inches and the height is \( 5\sqrt{3} \) inches.

Step2: Substitute the values into the formula

$$ A = \frac{1}{2} \times 4\sqrt{20} \times 5\sqrt{3} $$

Step3: Simplify the coefficients and the square roots

First, simplify \( \sqrt{20} \). We know that \( \sqrt{20}=\sqrt{4\times5} = 2\sqrt{5} \). So, substitute \( \sqrt{20} = 2\sqrt{5} \) into the expression:

$$ A=\frac{1}{2}\times4\times2\sqrt{5}\times5\sqrt{3} $$

Now, multiply the coefficients: \( \frac{1}{2}\times4\times2\times5=\frac{1}{2}\times40 = 20 \)
And multiply the square roots: \( \sqrt{5}\times\sqrt{3}=\sqrt{15} \)
So, \( A = 20\sqrt{15} \) square inches.

Answer:

\( 20\sqrt{15} \text{ in}^2 \)