Question

which set of additional details are the minimum required to ensure the new design will have the same volume as the original design?
a. the heights of both designs must be proportional, the base areas of both designs must be proportional, and each cross - section taken at the same height on both designs must have the same radius.
b. the heights of both designs must be the same, and at least one cross - section taken at the same height on both designs must have the same area.
c. the heights of both designs must be the same, and the base areas of both designs must be the same.
d. the heights of both designs must be the same, the base areas of both designs must be the same, and each cross - section taken at the same height on both designs must have the same area.

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is \(V = \pi r^{2}h\) (where \(r\) is the radius of the base and \(h\) is the height). For two cylinders (original and new design), if we want \(V_1=V_2\), and assume the height \(h\) is the same for both cylinders (since the problem is about additional details for volume - same - ness). The base area of a cylinder is \(A=\pi r^{2}\). If \(h_1 = h_2=h\), then \(V_1=\pi r_1^{2}h\) and \(V_2=\pi r_2^{2}h\). For \(V_1 = V_2\), we just need the base areas (\(\pi r_1^{2}\) and \(\pi r_2^{2}\)) to be the same (because \(h\) is non - zero). Also, consider the cross - section of a cylinder parallel to the base. The area of a cross - section parallel to the base of a cylinder is equal to the base area. If the heights are the same and the base areas (and thus cross - sectional areas parallel to the base) are the same, by the formula \(V=Ah\) (where \(A\) is the base area and \(h\) is the height), the volumes will be the same.

Step2: Analyze each option

• Option A: If the heights are proportional and base areas are proportional, \(V=\pi r^{2}h\), and if \(r_2 = kr_1\) and \(h_2=kh_1\), then \(V_2=\pi(kr_1)^{2}(kh_1)=k^{3}\pi r_1^{2}h_1=k^{3}V_1

eq V_1\) (unless \(k = 1\)).

• Option B: If the heights are the same and at least one cross - section (not parallel to the base) is the same, it doesn't guarantee volume equality. For example, if we take a slanted cross - section (not parallel to the base), it doesn't relate directly to the volume formula \(V=\pi r^{2}h\).
• Option C: If heights are the same and base areas are the same, using \(V=Ah\) (where \(A\) is the base area and \(h\) is the height), \(V_1 = V_2\). The cross - section parallel to the base has an area equal to the base area. Since \(h\) is the same and \(A\) is the same, the volume is the same.
• Option D: If only the height is the same, and no information about the base area (since \(V = Ah\)), if \(A\) (base area) is different, \(V\) will be different.

Answer:

C. The heights of both designs must be the same, and the base areas of both designs must be the same.