Big Ideas Math Geometry, 2014
BI
Big Ideas Math Geometry, 2014 View details
5. Volumes of Prisms and Cylinders
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Exercise 7 Page 630

Start by finding the area of the base of the given solid.

150sqrt(3)-45 cubic feet

Practice makes perfect

Let's consider the given composite solid.

We want to find the volume of this solid. First, let's find the area of its base B. To do this, we will subtract the area of the square from the area of the equilateral triangle.

We can find the area of the square by squaring its side length. A_(□)=(3 ft)^2 ⇔ A_(□)= 9 ft^2 To find the area of the triangle, we first need to find its height. To do so, we will consider the right triangle formed by one of the sides, the height, and half the base.

In this right triangle, the legs are 5 and h, and the hypotenuse is 10. We can find the value of h by using the Pythagorean Theorem.
a^2+b^2=c^2
5^2+ h^2= 10^2
â–Ľ
Solve for h
25+h^2=100
h^2=75
h=sqrt(75)
h=sqrt(25(3))
h=5sqrt(3)
The height of the right triangle, and also the height of the equilateral triangle, is 5sqrt(3) feet. With this information, we can find the area A_(â–ł).
A_(â–ł)=1/2bh
A_(â–ł)=1/2( 10)(5sqrt(3))
â–Ľ
Evaluate right-hand side
A_(â–ł)=10/2(5sqrt(3))
A_(â–ł)=5(5sqrt(3))
A_(â–ł)=25sqrt(3)
The area of the equilateral triangle is A_(â–ł)= 25sqrt(3) square feet. Therefore, we can find the area of the base B.
B= A_(â–ł)- A_(â–ˇ)
B= 25sqrt(3)- 9
This tells us that the area of the base is 25sqrt(3)-9 square feet. Now to get the volume of the composite solid, we will multiply the area of the base by the height of the solid 6 feet.
V=Base* height
V=(25sqrt(3)-9)(6)
V=150sqrt(3)-45
The volume of the given solid is 150sqrt(3)-45 cubic feet.