Question

the length of a rectangle is increasing at a rate of 7 in. / s, while its width is decreasing at 2 in. / s. as a result, the rectangles length \\(l\\), width \\(w\\), and its area \\(a\\) are all functions of time \\(t\\).

step 1 of 3: which equation describes the relationship between the rate of change of the rectangles area and the rates of change of its length and width?

• \\(\frac{da}{dt} = \frac{dl}{dt} \cdot w - l \cdot \frac{dw}{dt}\\)
• \\(\frac{da}{dt} = l \cdot w\\)
• \\(\frac{da}{dt} = \frac{dl}{dt} \cdot \frac{dw}{dt}\\)
• \\(\frac{da}{dt} = 2 \cdot \frac{dl}{dt} + 2 \cdot \frac{dw}{dt}\\)
• \\(\frac{da}{dt} = \frac{dl}{dt} \cdot w + l \cdot \frac{dw}{dt}\\)
• \\(\frac{da}{dt} = \frac{dl}{dt} + \frac{dw}{dt}\\)

Explanation:

State the area formula

Using the Product Rule Differentiation knowledge point

$$ A = l \cdot w $$

Differentiate with respect to time

Using the Product Rule Differentiation knowledge point

$$ \frac{dA}{dt} = \frac{dl}{dt} \cdot w + l \cdot \frac{dw}{dt} $$

Answer:

• (A) \(\frac{dA}{dt} = \frac{dl}{dt} \cdot w - l \cdot \frac{dw}{dt}\)
• (B) \(\frac{dA}{dt} = l \cdot w\)
• (C) \(\frac{dA}{dt} = \frac{dl}{dt} \cdot \frac{dw}{dt}\)
• (D) \(\frac{dA}{dt} = 2 \cdot \frac{dl}{dt} + 2 \cdot \frac{dw}{dt}\)
• (E) \(\frac{dA}{dt} = \frac{dl}{dt} \cdot w + l \cdot \frac{dw}{dt}\) (Correct answer)
• (F) \(\frac{dA}{dt} = \frac{dl}{dt} + \frac{dw}{dt}\)