A design team determines that acost-efficient way of manufacturingcylindrical containers for theirproducts is to have the volume, V,in cubic centimetres, modelled byV(x) = 9πx3 + 51πx2 + 88πx + 48π, wherex is an integer such that 2 ≤ x ≤ 8. Theheight, h, in centimetres, of each cylinderis a linear function given by h(x) = x + 3.a) Determine the quotient V(x)_h(x) andinterpret this result.b) Use your answer in part a) to expressthe volume of a container in the formπr2h.
Question
A design team determines that acost-efficient way of manufacturingcylindrical containers for theirproducts is to have the volume, V,in cubic centimetres, modelled byV(x) = 9πx3 + 51πx2 + 88πx + 48π, wherex is an integer such that 2 ≤ x ≤ 8. Theheight, h, in centimetres, of each cylinderis a linear function given by h(x) = x + 3.a) Determine the quotient V(x)_h(x) andinterpret this result.b) Use your answer in part a) to expressthe volume of a container in the formπr2h.
Solution
a) To determine the quotient V(x)/h(x), we simply divide the function V(x) by the function h(x):
V(x)/h(x) = (9πx^3 + 51πx^2 + 88πx + 48π) / (x + 3)
This quotient represents the volume of the cylinder divided by its height. In terms of the physical properties of the cylinder, this is equivalent to the cross-sectional area of the cylinder (i.e., the area of the circle that forms the base of the cylinder). This is because the volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the base and h is the height. Therefore, dividing the volume by the height gives the area of the base: A = πr^2.
b) To express the volume of the container in the form πr^2h, we can use the result from part a). We know that the cross-sectional area of the cylinder is given by V(x)/h(x), which is equivalent to πr^2. Therefore, we can express the volume as:
V = πr^2h
Substituting the expression for πr^2 from part a) gives:
V = (9πx^3 + 51πx^2 + 88πx + 48π) / (x + 3) * (x + 3)
Simplifying this expression gives the volume of the container in the form πr^2h:
V = 9πx^3 + 51πx^2 + 88πx + 48π
This is the volume of the container in terms of the radius and height.
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