Sometimes it is useful to be able to work backwards from the surface area or volume of an object to find an unknown dimension (for example height, length, width, or radius).
Suppose you are asked to design a crate in the shape of a square prism with a capacity of $1$1 m3 and a height of $0.5$0.5 m. What should the side length of the square base be?
Let $x$x denote the unknown side length of the square base.
Since the crate is a prism its volume is given by the formula
$\text{Volume }=\text{ Area of base }\times\text{ height}$Volume = Area of base × height
We known the volume is $1$1 m3 and the height is $0.5$0.5 m. Since the base is a square with side length $x$x, the area of the base is $A=x^2$A=x2.
Substituting these values into the formula above gives:
$1=x^2\times0.5$1=x2×0.5
Next, divide each side by $0.5$0.5 to isolate the unknown:
$x^2$x2 | $=$= | $\frac{1}{0.5}$10.5 |
$=$= | $2$2 |
The solutions to this equation are:
$x=\pm\sqrt{2}$x=±√2
Since $x$x is a length it must be a positive quantity, so we take the positive solution:
$x=\sqrt{2}\approx1.41$x=√2≈1.41 m
Therefore the side length of the square base of the crate should be approximately $1.41$1.41 m.
As you can see from the example above, working backwards from surface area or volume typically requires rearranging an area or volume formula. For reference, below is a summary of all such formulas that we have seen so far.
Prisms |
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Pyramids |
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Cylinders |
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Cones |
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Spheres |
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A ball has a surface area of $50.27$50.27 mm2.
Determine its radius, to the nearest integer.
This wild animal house is made out of plywood.
If the nesting box needs to have a volume of $129978$129978 cm3 and a height of $83$83 cm and front width of $54$54 cm, find the depth of the box.
The volume of the following tent is $4.64$4.64 m3.
Determine the height $h$h of the tent, in cm.