Melissa designed a sculpture in a which a cylinder-shaped section was removed from a cube.
Before painting the sculpture, Melissa wants to sand the surface where the cylinder section was removed. We want to find the surface area of the section.
Because the hollowed out part is a cylinder, the surface area of the section is the lateral area of the cylinder. In the diagram below we can see its dimensions.
The lateral surface of a cylinder is a rectangle. Its height is the same as the height of the cylinder. Its base is the same as the circumference of the circular base. Now, remember that the circumference of a circle with diameter d is π d. We can also show this in a diagram.
Let's multiply the dimensions. This will give us the formula for the lateral area.
A = (Ï€ d)(h)
Now we can substitute the values 5 for d and 10 for h in the formula and evaluate the expression. We will use 3.14 for π.
The surface area of the section equals about 157 cm^2.
Melissa has a can of spray paint that covers about 6500 square centimeters. We want to know if she can apply two coats of paint to the entire sculpture. This means finding the total surface area of the sculpture.
We can divide the surface of the sculpture into two parts, the inner and outer surfaces. The outer surface is the surface of the cube with two circular bases removed. The inner surface is the lateral surface of the cylinder.
First, let's find the area of the cube.
Surface Area of the Cube
Let's remember that the surface area of a cube is six times the area of its face. If a is the length of the edge of the cube, we can write its surface area as follows.
S.A. = 6a^2From the diagram we know that the edge of the cube a has a length of 10 centimeters. Let's substitute 10 for a in the expression and evaluate.
The surface area of the cube equals 600 cm^2. Now, let's move to finding the area of the circular bases. Once we find this value, we can subtract it from the surface area of the cube. This will give us the outer area of the sculpture. Let's do it!
Area of the Bases
The bases of the cylinder are circles. We know that the diameter of each circle is 5 centimeters. If we divide this number by 2, we will get the radius of the circles.
r &= 5 cm/2 [0.5em]
& = 2.5 cm
Let's recall the formula for the area of a circle with radius r.
A = π r^2
Now we can substitute 5 for r in the expression and evaluate.
The area of each circular base is about 19.625 cm^2. This means that the area of both bases is 2(19.625)=39.25 cm^2.
Surface Area of the Sculpture
The surface area of the sculpture is the sum of two areas, the outer area and the inner area. The outer surface area is the surface area of the cube decreased by the area of the circular bases. We can calculate it by subtracting the found values.
600-39.25=560.75 cm^2
The inner surface area is the lateral area of the cylinder. Notice that we found this value in Part A of the exercise.
157 cm^2
Last, we should add the values. This will give us the total surface area of the sculpture.
560.75+157=717.75 cm^2
If we spray coat the sculpture twice, we will use 2(717.75)=1435.5 cm^2 of spray paint. Because 1435.5< 6500, we can be sure that Melissa can apply two coats of spray paint to the entire sculpture.
The volume of a cube is the cube of its side length. If a is the length of the edge of the cube, then we can write its volume as follows.
V = a^3
We know that the edge of the cube has a length a of 10 centimeters. Let's substitute it into the expression and evaluate.
To calculate the volume of a cylinder, we multiply the area of its base B by its height h.
V=Bh
We calculated the area of the base of the cylinder in Part B of the exercise.
B=19.625 cm^2
The height of the cylinder is 10 cm. Let's substitute these values into the formula for the volume and evaluate.
The volume of the sculpture is the difference of the two volumes we found. Let's subtract them.
1000-196.25 = 803.75 cm^3
The volume of the sculpture equals 803.75 cm^3.
