a Before writing the expression for the ratio, find the surface areas and the volumes for the given figures.
B
b You can use the expressions that are written in Part A.
A
aPrism: 2b+ 4h/bh Cylinder: 2r+ 2h/rh
B
bPrism: 4/9 Cylinder: 4/9
Practice makes perfect
a We are asked to write expressions for the ratios of surface area to volume of the given figures. To do so, we will first find their surface areas and volume separately. Then we can write the expression for their ratios. Now let's begin with the square prism.
Square Prism
We will take a look at the given square prism.
To find the surface area we will find and add up the base areas and the lateral area. Notice that the bases are squares and the lateral area consists of four equal rectangles with the same width h and length b.
Surface Area of the Square Prism
Base Areas
Lateral Area
Surface Area
b^2+b^2= 2b^2
4hb
2b^2+ 4hb
Now, remember that the volume of a prism is the multiplication of its base area and height. Since we have the base area b^2 and the height h, we can write an expression to define the volume.
Volume of the Prism = b^2h
From here we will write the ratio of surface area to the volume for the square prism, then we will simplify the expression.
We will now apply the same process for the given cylinder. Let's take a look at it!
Notice that the bases are circles and the lateral area is a rectangle width h and length 2Ï€ r. With this in mind we will write their areas one at a time.
Surface Area of the Square Prism
Base Areas
Lateral Area
Surface Area
Ï€ r^2+Ï€ r^2= 2Ï€ r^2
2Ï€ rh
2Ï€ r^2+ 2Ï€ rh
Recall that the volume of a cylinder is the product of the base area π r^2 and its height h.
Volume of the Cylinder = π r^2h
From here we will write the ratio of surface area to volume for the cylinder, then we will simplify the expression.
b Now we will find the ratios of the surface area to volume for the given figures by using the given values. We will begin with finding the ratio for the given square prism.
Square Prism
We found the ratio of the surface area to the volume for the square prism in Part A.
Surface Area of the Prism/Volume of the Prism= 2b+ 4hbh
We are given that b is 12 feet and h is 18 feet, so we will substitute these values into the expression and find the ratio.
Surface Area of the Prism/Volume of the Prism=2b+ 4h/bh
We will proceed the same way for the cylinder, as well. We are given h=18 feet and r=6 feet. We will substitute these values into the expression we found for the cylinder in Part A.
Surface Area of the Cylinder/Volume of the Cylinder=2r+ 2h/rh
