Question

Question was provided via image upload.

Explanation:

Identify the given values and formula

The volume \(V\) of a right circular cone is given by:

$$V = \frac{1}{3}\pi x^2 y$$

where \(x\) is the radius and \(y\) is the height.
We are given:

• The volume \(V\) is a constant: \(V = 324\pi\text{ cm}^3\)
• The radius \(x = 6\)
• The height \(y = 27\)

We need to find the derivative \(\frac{dy}{dx}\) at this state.

Set up the relation

Since \(V\) is constant, we substitute the constant value into the volume equation:

$$324\pi = \frac{1}{3}\pi x^2 y$$

Divide both sides by \(\pi\):

$$324 = \frac{1}{3}x^2 y$$

Multiply both sides by 3:

$$972 = x^2 y$$

Differentiate implicitly with respect to \(x\)

Differentiate both sides of the equation \(x^2 y = 972\) with respect to \(x\) using the product rule:

$$\frac{d}{dx}(x^2 y) = \frac{d}{dx}(972)$$

$$2xy + x^2 \frac{dy}{dx} = 0$$

Solve for \(\frac{dy}{dx}\)

Isolate the derivative term:

$$x^2 \frac{dy}{dx} = -2xy$$

Since \(x
eq 0\), divide both sides by \(x^2\):

$$\frac{dy}{dx} = -\frac{2y}{x}$$

Substitute the given values

Substitute \(x = 6\) and \(y = 27\) into the expression:

$$\frac{dy}{dx} = -\frac{2(27)}{6}$$

$$\frac{dy}{dx} = -\frac{54}{6} = -9$$

Answer:

The volume of a right circular cone of radius \(x\) and height \(y\) is given by \(V = \frac{1}{3}\pi x^2 y\). Suppose that the volume of the cone is a constant \(324\pi\text{ cm}^3\). Find \(\frac{dy}{dx}\) when \(x = 6\) and \(y = 27\).

\(\frac{dy}{dx} =\) <blank>-9</blank>