Answer:
This equation is not possible, which means there is no value of h that satisfies it. Therefore, there is no solution for h in this case.
Explanation:
(a) To express Vs in terms of h and pi, we can write the equation for Vs as follows: V_s = kh^2 + lh^3 where k and l are constants.
Given that when h = 2, V_s = (800pi)/3, we can substitute these values into the equation to get:
(800pi)/3 = k(2^2) + l(2^3)
(800pi)/3 = 4k + 8l
Given that when h = 3, V_s = 576pi, we can substitute these values into the equation to get:
576pi = k(3^2) + l(3^3)
576pi = 9k + 27l
Now we have a system of equations:
4k + 8l = (800pi)/3
9k + 27l = 576pi
To solve this system, we can first multiply the first equation by 9 and the second equation by 4 to eliminate k:
36k + 72l = 2400pi/3
36k + 108l = 2304pi
Subtracting the first equation from the second equation, we get:
36k + 108l - (36k + 72l) = 2304pi - (2400pi)/3
36l = 2304pi - 800pi
36l = 1504pi
l = (1504pi)/36
l = 44pi/3
Substituting this value of l into the first equation, we can solve for k:
4k + 8(44pi/3) = (800pi)/3
4k + 352pi/3 = 800pi/3
4k = 448pi/3
k = (448pi/3)/4
k = (112pi)/3
Therefore, Vs = kh^2 + lh^3 can be written as: Vs = ((112pi)/3)h^2 + (44pi/3)h^3
(b) To express VT in terms of h and pi, we can use the fact that the volume of the entire hemisphere is given by (2/3)π(18^3), and the volume of S is Vs: VT = (2/3)π(18^3) - Vs
Substituting the expression for Vs from part (a) into this equation, we get: VT = (2/3)π(18^3) - ((112pi)/3)h^2 - (44pi/3)h^3
(c) (i) If the volumes of S and T are the same, then Vs = VT. We can equate the expressions for Vs and VT from parts (a) and (b) respectively, and solve for h: ((112pi)/3)h^2 + (44pi/3)h^3 = (2/3)π(18^3) - ((112pi)/3)h^2 - (44pi/3)h^3
Simplifying this equation, we get:
((112pi)/3)h^2 + (44pi/3)h^3 + ((112pi)/3)h^2 + (44pi/3)h^3 = (2/3)π(18^3)
Combining like terms, we have:
((224pi)/3)h^2 + (88pi/3)h^3 = (2/3)π(18^3)
Simplifying further, we get:
(224/3)h^2 + (88/3)h^3 = (2/3)(18^3)
(224/3)h^2 + (88/3)h^3 = 23328
(ii) It is given that the volume of W is 2088pi cm³. We can write an equation for the volume of W: VW = (2/3)π(18^3) - Vs - V_T 2088π = (2/3)π(18^3) - ((112pi)/3)h^2 - (44pi/3)h^3 - ((2/3)π(18^3) - ((112pi)/3)h^2 - (44pi/3)h^3)
Simplifying this equation, we get:
2088π = (2/3)π(18^3) - (2/3)π(18^3)
2088π = 0
This equation is not possible, which means there is no value of h that satisfies it. Therefore, there is no solution for h in this case.