Video Transcript
The volumes of two solids A and B
are in the ratio eight to 27. The surface area of shape B is 171
centimeters squared. Find the surface area of shape
A.
In order to solve this problem, we
need to consider the volume scale factor, the area scale factor, and the length
scale factor. The length scale factor is the cube
root of the volume scale factor and the area scale factor is calculated by squaring
the length scale factor. As our two solids are in the ratio
eight to 27, the volume scale factor is 27 over eight or twenty-seven eighths.
To calculate the volume of solid B,
we multiply the volume of solid A by twenty-seven eighths. Likewise, to calculate the volume
of solid A, we would divide the volume of solid B by twenty-seven eighths.
Our next step is to calculate the
length scale factor. This is the cube root of
twenty-seven eighths. This can be rewritten as the cube
root of 27 divided by the cube root of eight. This is because the cube root of A
over B is the same as the cube root of A divided by the cube root of B. The cube root of 27 is equal to
three, as three cubed equals 27. And likewise, the cube root of
eight is equal to two, as two cubed equals eight. Therefore, the length scale factor
is three over two or 1.5.
The final scale factor we need to
calculate is the area scale factor. We do this by squaring the length
scale factor, in this case three over two squared. This is equal to nine over four as
three squared is equal to nine and two squared is equal to four. We could use the decimal values of
these three scale factors if easier. The volume scale factor is 3.375,
the length scale factor is 1.5, and the area scale factor is 2.25.
Our question told us that the
surface area of shape B is 171 centimeters squared. We need to calculate the surface
area of shape A. In order to do this, we’ll use the
area scale factor nine over four or 2.25. As shape A is smaller than shape B,
we need to divide by the area scale factor.
In this case, we need to divide 171
by nine over four or 2.25. We could just type this into the
calculator or we could use our method for dividing fractions. Dividing by a fraction is the same
as multiplying by the reciprocal of this fraction. Dividing by nine over four is the
same as multiplying by four over nine.
Nine divides into 171 19 times. Therefore, this is the same as 19
multiplied by four, which is equal to 76. This means that the surface area of
shape A is 76 centimeters squared. We would get the same answer if we
divided 171 by 2.25.
