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

Substitute base \( = 4\sqrt{20} \) and height \( = 5\sqrt{3} \) into the formula:

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

Step3: Simplify the expression

First, multiply the coefficients and the square roots separately.
Multiply the coefficients: \( \frac{1}{2}\times4\times5 = 10 \)
Multiply the square roots: \( \sqrt{20}\times\sqrt{3}=\sqrt{20\times3}=\sqrt{60} \)
Simplify \( \sqrt{60} \): \( \sqrt{60}=\sqrt{4\times15}=2\sqrt{15} \)
Now, multiply the result from the coefficients and the simplified square root: \( 10\times2\sqrt{15}=20\sqrt{15} \)

Wait, let's re - do the simplification of \( \sqrt{20} \) first. \( \sqrt{20}=\sqrt{4\times5} = 2\sqrt{5} \)
So, going back to the area formula:

$$ A=\frac{1}{2}\times(4\times2\sqrt{5})\times(5\sqrt{3})=\frac{1}{2}\times(8\sqrt{5})\times(5\sqrt{3}) $$

Multiply the coefficients: \( \frac{1}{2}\times8\times5=20 \)
Multiply the square roots: \( \sqrt{5}\times\sqrt{3}=\sqrt{15} \)
So, \( A = 20\sqrt{15} \) square inches? Wait, no, let's do it step by step correctly.

Wait, \( 4\sqrt{20}=4\sqrt{4\times5}=4\times2\sqrt{5}=8\sqrt{5} \)
Then, \( \frac{1}{2}\times8\sqrt{5}\times5\sqrt{3}=( \frac{1}{2}\times8\times5)\times(\sqrt{5}\times\sqrt{3}) \)
\( \frac{1}{2}\times8\times5 = 20 \)
\( \sqrt{5}\times\sqrt{3}=\sqrt{15} \)
So, \( A = 20\sqrt{15} \)? Wait, no, wait, let's check again.

Wait, original base: \( 4\sqrt{20} \), height: \( 5\sqrt{3} \)

\( \frac{1}{2}\times4\sqrt{20}\times5\sqrt{3}=\frac{1}{2}\times4\times5\times\sqrt{20}\times\sqrt{3} \)

\( \frac{1}{2}\times4\times5 = 10 \)

\( \sqrt{20}\times\sqrt{3}=\sqrt{60}=\sqrt{4\times15}=2\sqrt{15} \)

Then \( 10\times2\sqrt{15}=20\sqrt{15} \). Wait, but let's check the first way of simplifying \( \sqrt{20} \) as \( 2\sqrt{5} \):

\( 4\sqrt{20}=4\times2\sqrt{5}=8\sqrt{5} \)

\( \frac{1}{2}\times8\sqrt{5}\times5\sqrt{3}=(4\sqrt{5})\times(5\sqrt{3}) = 20\sqrt{15} \). Yes, that's correct.

Wait, another way:

\( \frac{1}{2}\times4\sqrt{20}\times5\sqrt{3}=\frac{1}{2}\times4\times5\times\sqrt{20}\times\sqrt{3} \)

\( 10\times\sqrt{60}=10\times\sqrt{4\times15}=10\times2\sqrt{15}=20\sqrt{15} \)

Answer:

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