A Canned Drink Will B E Made Using 40 In2 Of Aluminum. Letr B…
A canned drink will b e made using 40 in2 of aluminum. Letr be the radius of the can and let h be the height. Define a function in terms of r that can represent the volume of the can.
Step-by-step explanation:
A canned drink is a closed cylindrical figure. Recall that the total surface area of a closed cylinder is given by
[tex] SA = 2\pi rh + 2\pi r^2 [/tex]
Given that [tex] SA = 40\ in²,[/tex] we can express [tex] h [/tex] in terms of [tex] r. [/tex]
[tex] 40 = 2\pi rh + 2\pi r^2 \\ 20 = \pi rh + \pi r^2 \\ 20 = \pi r (h + r) \\ \dfrac{20}{\pi r} = h + r \\ h = \dfrac{20}{\pi r} – r [/tex]
We already know that the volume of a cylinder is given by
[tex] V = \pi r^2 h [/tex]
Define [tex] V(r) [/tex] as a function of [tex] r [/tex] that represents the volume of the can. Substituting [tex] h[/tex] to the formula for volume, we get
[tex] V = \pi r^2 h \\ V(r) = \pi r^2\left(\dfrac{20}{\pi r} – r\right) \\ \boxed{V(r) = 20r – \pi r^3} [/tex]
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