The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 7 cm/s. When the length is 15 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
A = lw
dA/dt = l dw/dt + w dl/dt
for the given case
dA/dt = 15(9) + 10(7)
= ...
To find how fast the area of the rectangle is increasing, we can use the formula for the rate of change of area:
Rate of change of area = Rate of change of length * Width + Length * Rate of change of width
Given:
Rate of change of length = 9 cm/s
Rate of change of width = 7 cm/s
Length = 15 cm
Width = 10 cm
Substituting the given values into the formula, we get:
Rate of change of area = 9 cm/s * 10 cm + 15 cm * 7 cm/s
Calculating the values, we have:
Rate of change of area = 90 cm²/s + 105 cm²/s
Rate of change of area = 195 cm²/s
Therefore, the area of the rectangle is increasing at a rate of 195 cm²/s.
To find how fast the area of the rectangle is increasing, we can use the formula for the rate of change of the area with respect to time, which is given by:
Rate of change of area = (rate of change of length) × (width) + (length) × (rate of change of width)
Given that the rate of change of length is 9 cm/s, the rate of change of width is 7 cm/s, the length is 15 cm, and the width is 10 cm, we can substitute these values into the formula:
Rate of change of area = (9 cm/s) × (10 cm) + (15 cm) × (7 cm/s)
Simplifying, we get:
Rate of change of area = 90 cm²/s + 105 cm²/s
Adding these values together, we get:
Rate of change of area = 195 cm²/s
Therefore, the area of the rectangle is increasing at a rate of 195 cm²/s.