We are given that a sink with a sliding lid is in the shape of the following rectangular prism.
We want to find the volume and surface area of a second sink knowing that it has a similar shape and is smaller by a scale factor of 12. Let's start by finding the volume and surface area of the larger sink. Recall that the volume and surface area of a rectangular prism with length l, width w, and height h can be calculated using the following formulas.
V & = l w h [0.3em]
S.A. & = 2( l w + w h + l h)In this case, l = 16, w = 15, and h= 8. We can substitute these values into the formulas to calculate the volume and surface area of the larger sink. We will start with the volume.
Therefore, the volume of the larger sink is 1920 cubic inches and the surface area is 976 square inches. Now we can find the volume and surface area of the smaller sink using the fact that it is smaller by a scale factor of 12. To do so, let's recall the relationships between the volumes and the surface areas of similar solids.
Volume of Similar Solids
If Solid X is similar to Solid Y by a scale factor, then the volume of X is equal to the volume of Y times the cube of the scale factor.
Surface Area of Similar Solids
If Solid X is similar to Solid Y by a scale factor, then the surface area of X is equal to the surface area of Y times the square of the scale factor.
We know that the two sinks are similar and the scale factor is 12. This means that we can calculate the volume and surface area of the smaller sink using these relationships. Let's do it!
Object
Volume, in^3
Surface Area, in^2
larger sink
1920
976
smaller sink
1920 (1/2)^3 = 240
976 (1/2)^2 = 244
Therefore, the volume of the smaller sink is 240 cubic inches and the surface area is 244 square inches.
