Variations?

哈哈，近來很多 rate, ratio, variation 的題目啊～
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(a)
C = k₁V + k₂A where k₁ and k₂ are non-zero constants.
Denote r and h respectively the radius and height of a cylinder, both in cm.
Note that the volume of a cylinder is V = π r² h.
The surface area of a cylinder is A = 2 π r² + 2 π r h.
Therefore,
C = k₁ π r² h + k₂ (2 π r² + 2 π r h)
C = k₁ π r² h + k₂ 2 π r (r + h)
The given information implies
{ 600π = k₁ π 2² × 10 + k₂ 2 π × 2 (2 + 10)
{ 615π = k₁ π 3² × 5 + k₂ 2 π × 3 (3 + 5)

{ 600 = 40 k₁ + 48 k₂
{ 615 = 45 k₁ + 48 k₂

5 k₁ = 15 ⇒ k₁ = 3

600 = 40 (3) + 48 k₂
48 k₂ = 480 ⇒ k₂ = 10

Therefore,
C = 3V + 10A.

(b)
Consider
C = 3V + 10A
C = 3 π r² h + (10) 2 π r (r + h)
C = 3 π r² h + 20 π r (r + h)
1000 = 3 π r² (8) + 20 π r (r + 8)
1000 = 24 π r² + 20 π r² + 160 π r
1000 = 44 π r² + 160 π r
250 = 11 π r² + 40 π r
11 π r² + 40 π r - 250 = 0　〔in the form of ar² + br + c = 0〕
r = 1.42837031 or -5.064733946 (rejected)
The required radius is 1.42837031 cm.