Question

Surface Area All edges of a cube are expanding at a rate of 8 centimeters per second. How fast is the surface area changing when each edge is 6.5 centimeters?

Answer 1

The surface area of the cube is increasing at a rate of 624 cm²/s when each edge is 6.5 cm, with the edges expanding at 8 cm/s.

Step-by-step explanation:

The question is asking for the rate of change of the surface area of a cube where each edge is expanding at a rate of 8 centimeters per second. The formula for the surface area of a cube is SA = 6s², where s is the length of a side of the cube. When s is changing, we can find the rate of change of the surface area by taking the derivative with respect to time (t), which gives us d(SA)/dt = 6 * 2s * ds/dt. Given that ds/dt (the rate of change of the side length) is 8 cm/s and s (the side at the instance we are considering) is 6.5 cm, we can substitute these values into the derivative to find the rate of change of the surface area.

So we have:

d(SA)/dt = 6 * 2 * 6.5 cm * 8 cm/s = 6 * 13 cm * 8 cm/s = 624 cm²/s. Therefore, the surface area of the cube is increasing at a rate of 624 square centimeters per second when each edge is 6.5 centimeters.

Answer 2

The surface area of the cube is increasing at a rate of 624 square centimeters per second when each edge is 6.5 centimeters long.

Step-by-step explanation:

The student is asking how fast the surface area of a cube is changing when each edge is increasing in length at a constant rate. Since all edges of the cube are expanding at a rate of 8 centimeters per second, we can use this information to calculate how quickly the surface area is changing. The surface area (SA) of a cube is given by the formula SA = 6s², where s is the length of one side of the cube.

If the rate of change of the side length (ds/dt) is 8 cm/s, we can find the rate of change of the surface area (dSA/dt) by differentiating the surface area formula with respect to time. Using the chain rule, we have:

dSA/dt = 6 × 2s × (ds/dt) = 12s × (ds/dt)

When each side is 6.5 cm:

dSA/dt = 12 × 6.5 cm × 8 cm/s

dSA/dt = 12 × 52 cm²/s

dSA/dt = 624 cm²/s

So, the surface area of the cube is increasing at a rate of 624 square centimeters per second when each edge is 6.5 centimeters long.