a Remember that the bottom surface of the cake will not be frosted, so you do not need to calculate its area.
B
b Remember that the bottom surface of the cake will not be frosted, so you do not need to calculate its area.
C
c Divide the surface area of each cake by 50.
D
d Find the surface area of the rectangular prism cake with the new height. The surface area of the cylindrical cake is the same.
A
a 96 in^2
B
b 113.1 in^2
C
cRectangular prism cake: 2 cans
Cylindrical cake: 3 cans
D
d r=2.18 in
Practice makes perfect
a We can calculate the surface area of a prism using the formula below, where P is the perimeter of the base, h is the height, and B is the area of the base. Note that we will not be using the full formula for surface area of a rectangular prism, because the bottom side of the cake will not be frosted.
S(including the bottom Base)=Ph+ 2B
S(excluding the bottom Base)=Ph+ B
Let's first calculate the perimeter and area of the base.
Perimeter of the Base
Since the base is a rectangle, we can calculate its perimeter by using the formula.
P=2(w+l)
Here w is the width and l is the length of the rectangle. Let's substitute 4 for l and 3 for w into the formula and solve for P.
We are told that each layer of the cake is 3 inches high. From the diagram we can see that there are 2 layers in the cake. Therefore, the total height of the cake is
h=3* 2=6in.
Now we know everything we need to calculate the surface area of the cake that is going to be frosted. Let's substitute P with 14, B with 12, and h with 6 into the formula.
The surface area of the cake that will be frosted is 96in^2.
b To calculate the surface area of the cylindrical cake, we can use the formula below.
S=2π rh+2π r^2
Here r is the radius of the base and h is the height of the cylinder. The bottom of the cake will not be frosted, so we should subtract its area. In the formula, the second term 2π r^2 represents the area of the two cylinder's bases. Since we want to calculate only the surface of the top, we can write the formula as π r^2 instead of 2π r^2.
S_(frosted)=2π rh+π r^2
To calculate the surface area we need to know the radius and the height. From the diagram we know that the radius of the base is 2 inches. It is also given that each layer of the cake is 4 inches in height. There are 2 layers in the cake, so the total height is
h=2* 4=8inches.
Now, let's substitute these values onto the formula and solve for S_(frosted).
The area of the cake's surface that will be frosted is approximately 113.1in^2.
c Let's find the number of the cans of frosting for each cake separately.
Rectangular Prism Cake
In Part A we found that the surface area of the rectangular prism cake that will be frosted is 96in^2. It is given that one can of frosting covers 50in^2 of cake. Let's divide 96 by 50 to find how many cans will be needed to cover the whole cake.
96/50=1.92 ≈ 2cans
Because we cannot buy parts of cans, 2 cans of frosting will be needed to cover the rectangular prism cake.
Cylindrical Cake
In Part B we calculated that the surface area of the cylindrical cake that is going to be frosted is 113.1in^2. Again, dividing 113.1 by 2 we can find the number of cans that are needed to frost the cake.
113.1/50=2.262
Because we cannot buy parts of cans, 3 cans of frosting will be needed to frost the cylindrical cake.
d Let's calculate the height of the rectangular prism cake if each layer of cake is 5 inches.
h=2* 5=10inches.
Now let's calculate the surface area of the rectangular prism cake that will be frosted by substituting the new height in the formula from Part A. Notice that the rest of the dimensions are left without changes. Thus, we can substitute P with 14, B with 12, and h with 10 into the formula.
Therefore, the surface area of the cake that will be frosted is 152in^2. Since the same amount of the frosting will be used for the cylindrical cake, it's surface area is also 152in^2. Let's substitute S_(frosted) with 152 and h with 10 into the formula from Part B.
152=2π r( 10)+π r^2
This way we get the equation for the unknown r. If we solve it, we will find the radius that the cylindrical cake should have for the surface area to be 152in^2. Let's do it!
We arrived at the quadratic equation. In order to solve it, we can use the Quadratic Formula.
ax^2+ bx+ c=0 ⇔ x=- b± sqrt(b^2-4 a c)/2 a
Let's compare our equation with the general quadratic equation above and find the coefficients a, b, and c.
ax^2+ bx+ c=0
1* r^2+ 20r+( - 48.38)=0
Thus, in our case a is 1, b is 20, and c is - 48.38. Let's substitute these values into the formula and find the solution of the equation.
Let's now find two different solutions of the equation by splitting the final fraction into the positive and negative cases.
r=- 20± 24.36/2
r=- 20+ 24.36/2
r=- 20- 24.36/2
r=4.36/2
r=- 44.36/2
r=2.18
r=- 22.18
Since the radius cannot be negative, the second solution is not suitable for the answer. Therefore, if the radius of the cylindrical cake is about 2.18 in, the amount of frosting used for each cake will be the same.
