With a given volume (V = 1800in²) and a given scale factor (SF or k = 2), one might be inclined to simply multiply the given volume by two as the previous responses have illustrated in their explanations (V’ = 3600in²). But let us examine scale factor a bit closer as it is applied to area and volume:
Scale factor is defined as a “ratio or number which is used to change the size of a figure without changing its shape.” It is used to increase (k>1) or decrease (k<1, a fraction) the size of an object. Scale factor is applied to linear dimensions.
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For one-dimensional figures (a line), yes, simply multiply the line length (L) by scale factor k = 2.
ie. L=6”,
L’ = L•k = 6”•2 = 12”
Our scale factor (k) has DOUBLED the given length.
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For two-dimensional figures (a rectangle, L•W), we multiply EACH dimension by scale factor k = 2.
ie. L=3”, W=5”
L’ = L•k = 3”•2 = 6”
W’ = W•k = 5”•2 = 10”
Our scale factor (k) has DOUBLED the given lengths.
Let’s see how scale factor impacts AREA of this example…
A = L•W =3 ”•5” = 15in²
A’ = L’•W’ = 6”•10” = 60in²
Our AREA scale factor = A’/A = 60in² / 15in² = 4, or FOUR times (k²) the original area!
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For three-dimensional figures (a prism, L•W•D), we multiply EACH dimension by scale factor k = 2.
ie. L=3”, W=5”, D=6”
L’ = L•k = 3”•2 = 6”
W’ = W•k = 5”•2 = 10”
D’ = D•k = 6”•2 = 12”
Our scale factor (k) has DOUBLED the given lengths.
Let’s see how scale factor impacts VOLUME of this example…
V = L•W•D = 3”•5”•6” = 90in³
V’ = L’•W’•D’ = 6”•10”•12” = 720in³
Our volume scale factor = V’/V = 720in² / 90in² = 8, or EIGHT times (k³) the original volume!!
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In summary…
•Linear, multiply by k
•Area, multiply by k²
•Volume, multiply by k³
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In the given problem, V = 1800in³, k =2. So…
V’ = V•k³
=1800in³•2³
=1800in³•8
∴V’ = 14,400in³…8times increase in the given volume!!
A prism, with dimensions (L•W•D) equal to 15”•12”•10”, yields a volume of 1800in³…our given volume. Try applying the scale factor k=2 to EACH of these dimensions, determine the new volume, and calculate the VOLUME scale factor (k³ = V’/V). Then solve for “k”.
