a grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. the metal…

a grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. the metal unit of surface area). if the volume of the silo is 950 m³, what are the dimensions of the silo (radius and hei the materials? assume the silo has no floor and no flat ceiling under the dome. what is the function of the cost of the silo, c, in terms of the radius, r? c = (9.2π/3)r² + 1900/r (type an expression. type an exact answer, using π as needed.) the interval of interest of the objective function is (0,7.7. (simplify your answer. type your answer in interval notation. round to the nearest tenth as needed.) the radius of the cylindrical base (and of the hemispherical dome) is (round to the nearest tenth as needed.)

a grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. the metal unit of surface area). if the volume of the silo is 950 m³, what are the dimensions of the silo (radius and hei the materials? assume the silo has no floor and no flat ceiling under the dome. what is the function of the cost of the silo, c, in terms of the radius, r? c = (9.2π/3)r² + 1900/r (type an expression. type an exact answer, using π as needed.) the interval of interest of the objective function is (0,7.7. (simplify your answer. type your answer in interval notation. round to the nearest tenth as needed.) the radius of the cylindrical base (and of the hemispherical dome) is (round to the nearest tenth as needed.)

Answer

Explanation:

Step1: Recall cost - function and find its derivative

We have the cost function $C(r)=\frac{9.2\pi}{3}r^{2}+\frac{1900}{r}$. Using the power - rule for differentiation, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. The derivative $C^\prime(r)=\frac{18.4\pi}{3}r-\frac{1900}{r^{2}}$.

Step2: Set the derivative equal to zero

To find the critical points, we set $C^\prime(r) = 0$. So, $\frac{18.4\pi}{3}r-\frac{1900}{r^{2}}=0$. Multiply through by $r^{2}$ to get $\frac{18.4\pi}{3}r^{3}-1900 = 0$. Then, $\frac{18.4\pi}{3}r^{3}=1900$.

Step3: Solve for $r$

First, isolate $r^{3}$: $r^{3}=\frac{1900\times3}{18.4\pi}$. Calculate $\frac{1900\times3}{18.4\pi}=\frac{5700}{18.4\pi}\approx98.07$. Then, $r=\sqrt[3]{98.07}\approx4.6$.

Answer:

$4.6$