Question

1. multiple choice if the volume of a cube is increasing at 24 in³/min and each edge of the cube is increasing at 2 in./min, what is the length of each edge of the cube? (a) 2 in. (b) 2√2 in. (c) ∛12 in (d) 4 in. (e) 8 in. 39. multiple choice if the volume of a cube is increasing at 24 in³/min and the surface area of the cube is increasing at 12 in²/min, what is the length of each edge of the cube? (a) 2 in. (b) 2√2 in. (c) ∛12 in. (d) 4 in. (e) 8 in.

Explanation:

: Recall volume formula for cube
The volume formula of a cube is $V = x^{3}$, where $x$ is the length of the edge. Differentiate with respect to time $t$ using the chain - rule. $\frac{dV}{dt}=3x^{2}\frac{dx}{dt}$.

Step2: Substitute given values

We know that $\frac{dV}{dt} = 24$ in³/min and $\frac{dx}{dt}=2$ in/min. Substitute these values into the equation $\frac{dV}{dt}=3x^{2}\frac{dx}{dt}$:
$24 = 3x^{2}\times2$.

Step3: Solve for $x$

First, simplify the right - hand side of the equation: $3x^{2}\times2 = 6x^{2}$. So, $6x^{2}=24$. Then divide both sides by 6: $x^{2}=4$. Take the square root of both sides, and since $x>0$ (as it represents a length), $x = 2$ in.

Step1: Recall volume and surface - area formulas for cube

The volume formula of a cube is $V=x^{3}$ and the surface - area formula is $S = 6x^{2}$. Differentiate both with respect to time $t$ using the chain - rule. $\frac{dV}{dt}=3x^{2}\frac{dx}{dt}$ and $\frac{dS}{dt}=12x\frac{dx}{dt}$.

Step2: Use the given values of $\frac{dV}{dt}$ and $\frac{dS}{dt}$

We know that $\frac{dV}{dt}=24$ in³/min and $\frac{dS}{dt}=12$ in²/min. From $\frac{dS}{dt}=12x\frac{dx}{dt}$, we can express $\frac{dx}{dt}=\frac{\frac{dS}{dt}}{12x}$. Substitute $\frac{dS}{dt} = 12$ into it, we get $\frac{dx}{dt}=\frac{12}{12x}=\frac{1}{x}$.

Step3: Substitute into the volume - rate equation

Substitute $\frac{dV}{dt}=24$ and $\frac{dx}{dt}=\frac{1}{x}$ into $\frac{dV}{dt}=3x^{2}\frac{dx}{dt}$. We have $24=3x^{2}\times\frac{1}{x}$.

Step4: Simplify and solve for $x$

Simplify the right - hand side: $3x^{2}\times\frac{1}{x}=3x$. So, $3x = 24$, and then $x = 8$ in.

Answer:

A. 2 in.

39.