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Example 1
The volume $V$ of a cylindrical tube with circular base is given by by $V=\pi r^2 \times h$ where $\pi$ is a constant and $r$ is the radius of the base and $h$ is the height of the tube. Which equation may be used to find the radius $r$ of a cylindrical tube whose height is $2$ cm less than $4$ times its radius and its volume is $4000$ cm$^3$?
- $r^2 \times (4 r - 2) = \dfrac{4000}{\pi}$
- $4 r^3 = \dfrac{4000}{\pi}$
- $4 r^3-2= \dfrac{4000}{\pi}$
- $8 \pi r = 4000$
- $4 r^3+2= \dfrac{4000}{\pi}$
Solution
- The given formula for the volume $V=\pi r^2 \times h$ can be used as a starting point since it has the form of an equation with many unknown. Let us first start by substituting the volume $V$ by its given value $4000$ in the formula
$4000=\pi r^2 \times h$
- We need an equation in $r$ but for the time being there two unknown: $r$ and $h$. However there is a relationship between $r$ and $h$: "whose height is $2$ cm less than $4$ times its radius" which may be translated mathematically as:
$h=4 r-2$
- We now substitute $h$ by $4 r-2$ in the equation $4000=\pi r^2 \times h$ to obtain an equation with one unknown $r$.
$4000=\pi r^2 \times (4 r -2)$
- Divide both sides of the above equation by $\pi$to obtain
$\dfrac{4000}{\pi}=r^2 \times (4 r -2)$
Answer A
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