The question involves finding the height of a cylindrical vase using given functions for the vase's capacity [tex]\(C(x)\)[/tex] and the base area [tex]\(A(x)\)[/tex].
Given:
- The capacity [tex]\(C(x)\)[/tex] of the vase is modeled by the function [tex]\(C(x) = 6.28 x^3 + 28.26 x^2\)[/tex], where [tex]\(x\)[/tex] is the radius of the vase in centimeters.
- The area of the circular base [tex]\(A(x)\)[/tex] of the vase is modeled by the function [tex]\(A(x) = 3.14 x^2\)[/tex].
To find the height [tex]\(h\)[/tex] of the vase, we use the formula for the volume of a cylinder:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Rewriting this formula in terms of the given functions:
[tex]\[ C(x) = A(x) \times h \][/tex]
Solving for [tex]\(h\)[/tex]:
[tex]\[ h = \frac{C(x)}{A(x)} \][/tex]
Substituting the given functions:
[tex]\[ h = \frac{6.28 x^3 + 28.26 x^2}{3.14 x^2} \][/tex]
Now let's simplify this expression:
[tex]\[ h = \frac{6.28 x^3 + 28.26 x^2}{3.14 x^2} = \frac{6.28 x^3}{3.14 x^2} + \frac{28.26 x^2}{3.14 x^2} \][/tex]
[tex]\[ h = 2x + 9 \][/tex]
By assigning [tex]\(x\)[/tex] (a particular positive value for radius), let's determine the height. Given the value [tex]\(x = 2\)[/tex]:
1. Calculate the capacity [tex]\(C(2)\)[/tex]:
[tex]\[ C(2) = 6.28 (2)^3 + 28.26 (2)^2 \][/tex]
[tex]\[ C(2) = 6.28 \cdot 8 + 28.26 \cdot 4 \][/tex]
[tex]\[ C(2) = 50.24 + 113.04 \][/tex]
[tex]\[ C(2) = 163.28 \][/tex]
2. Calculate the base area [tex]\(A(2)\)[/tex]:
[tex]\[ A(2) = 3.14 (2)^2 \][/tex]
[tex]\[ A(2) = 3.14 \cdot 4 \][/tex]
[tex]\[ A(2) = 12.56 \][/tex]
3. Finally, calculate the height [tex]\(h\)[/tex]:
[tex]\[ h = \frac{C(2)}{A(2)} = \frac{163.28}{12.56} \][/tex]
[tex]\[ h = 13 \][/tex]
So, the height of the vase is 13 centimeters when the radius is 2 centimeters.
