c Take the difference between the volume of the outside and inside prisms.
A
a 1044.1 cm^2
B
b 553.6 cm^2
C
c 1481.4 cm^3
Practice makes perfect
a To find the surface area of the exterior of the box, let's first recall the formula for the surface area of a prism.
S.A= p h+2 B, where p is the perimeter of the base and B is the area of the base
We are given a height of 14 centimeters. Let's start by finding the perimeter of the base. The base is a pentagon, which means it has five sides. We are given the length of the base edge, so the perimeter will be five times this.
Exterior p=5 * (Base edge)
⇕
Exterior p=5 * (10 cm)= 50 cm
The perimeter of the base is 50 centimeters. To find the area of the base, let's recall the formula for the area of a pentagon.
Area of a pentagon=1/4sqrt(5*(5+2sqrt(5)))s^2, where s is the length of the base edge
Let's substitute the base edge of 10 centimeters to find the area of the base.
The outside surface area of the box is about 1044.1 square centimeters.
b To find the inside surface area, we will repeat the same process as Part A. We are given a height of 11 centimeters. Let's use the length of the base edge of 7 centimeters to find the perimeter.
Interior p=5 * (Base edge)
⇕
Interior p=5 * (7 cm)= 35 cmThe perimeter of the base is 35 centimeters. Let's substitute the base edge into the formula for the area of a pentagon to find the area of the base.
The inside surface area of the box is about 553.6 square centimeters.
c The volume of the material needed to make the box can be found by taking the difference of the volume of the outside and inside prisms. Let's recall the formula for the volume of a prism.
V= B h, where B is the area of the baseWe already found the area of the bases for the outside and inside prisms in Parts A and B, respectively. Let's substitute these values and the given heights to find the volume of each prism.
Prism
Area of Base
Height
V=Bh
Outside
172.05 cm^2
14 cm
2408.7 cm^3
Inside
84.3 cm^2
11 cm
927.3 cm^3
The volume of the outside prism is 2408.7 cubic centimeters. The volume of the inside prism is 927.3 cubic centimeters. Finally, we will take the differences of these volumes to find the volume of material needed to make the box.
Volume of material needed=V_(Outside)-V_(Inside)
⇕
Volume of material needed=2408.7-927.3=1481.4 cm^3
The volume of material needed to make the box is 1481.4 cubic centimeters.
