the length of a rectangular prism is increasing at a rate of 8 cm/s, its width is increasing at a rate of 3 cm/s, and its height is increasing at a rate of 5 cm/s. when the length is 20 cm, width is 10 cm, and height is 15 cm, how fast is the volume of the rectangular prism increasing?
V = lwh
dV/dt = lw dh/dt + lh dw/dt + wh dl/dt
you are given all the values, just plug in and evaluate
To find out how fast the volume of the rectangular prism is increasing, we can use the formula for the volume of a rectangular prism:
Volume = Length × Width × Height.
Given that the length is increasing at a rate of 8 cm/s, the width is increasing at a rate of 3 cm/s, and the height is increasing at a rate of 5 cm/s, we need to find the rate of change for the volume.
Let's call the length, width, and height of the rectangular prism at any given time as L(t), W(t), and H(t), respectively. We are given that L(0) = 20 cm, W(0) = 10 cm, and H(0) = 15 cm.
Differentiating both sides of the volume formula with respect to time, t, using the chain rule:
d/dt [Volume] = d/dt [L(t) × W(t) × H(t)].
Now, let's differentiate each term:
d/dt [Volume] = d/dt [L(t)] × W(t) × H(t) + L(t) × d/dt [W(t)] × H(t) + L(t) × W(t) × d/dt [H(t)].
Since L(t) = 20, W(t) = 10, and H(t) = 15, we substitute these values.
d/dt [Volume] = 8 × 10 × 15 + 20 × 3 × 15 + 20 × 10 × 5.
Simplifying:
d/dt [Volume] = 1200 + 900 + 1000.
d/dt [Volume] = 3100 cm³/s.
Therefore, the volume of the rectangular prism is increasing at a rate of 3100 cm³/s.