Part B: This bath time rubber ducky has a surface area of 84 cm2 and a volume of 486 cm3. The bathtime ducky and the
pool float ducky are similar figures. If all of the dimensions of the bath time rubber ducky are multiplied by 16 to get this
pool float rubber ducky, what will the surface area and the volume of the pool float ducky be?
Let's use the fact that similar figures have proportional side lengths. If we multiply all dimensions of the bath time rubber ducky by 16, we'll get the dimensions of the pool float rubber ducky.
First, let's find the dimensions of the bath time rubber ducky. We have:
Surface area = 84 cm^2
Volume = 486 cm^3
Let's label the dimensions of the rubber ducky as follows:
Length = l
Width = w
Height = h
We know that the surface area is given by:
Surface area = 2lw + 2lh + 2wh
Substituting in the values we have:
84 = 2lw + 2lh + 2wh
And we also know that volume is given by:
Volume = lwh
Substituting in the values we have:
486 = lwh
We have two equations and three unknowns (l, w, and h), so we can't solve for them directly. However, we do know that the rubber ducky is a similar figure to the pool float rubber ducky, so we can use the proportionality of side lengths to find the dimensions of the pool float rubber ducky.
If we multiply all dimensions of the bath time rubber ducky by 16 to get the pool float rubber ducky, we get:
Length = 16l
Width = 16w
Height = 16h
We can use these proportions to find the surface area and volume of the pool float rubber ducky.
Surface area:
The surface area of the pool float rubber ducky is given by:
Surface area = 2(16l)(16w) + 2(16l)(16h) + 2(16w)(16h)
This simplifies to:
Surface area = 2048lw + 2048lh + 2048wh
We can substitute in the values we have for the bath time rubber ducky:
Surface area = 2048(l)(w) + 2048(l)(h) + 2048(w)(h)
And we know that lw = 84/2 - lh = 486/(lw) from the bath time rubber ducky dimensions:
Surface area = 2048(84/2lw) + 2048(486/lw) + 2048(84/2lh)
Simplifying, we get:
Surface area = 86,016/ lw
So the surface area of the pool float rubber ducky is 86,016 cm^2.
Volume:
The volume of the pool float rubber ducky is given by:
Volume = (16l)(16w)(16h)
This simplifies to:
Volume = 4096lwh
We can substitute in the values we have for the bath time rubber ducky:
Volume = 4096(486)
So the volume of the pool float rubber ducky is 1,993,216 cm^3.