Question
Suppose that a dimension \( x \) and the area \( A=11 x^{2} \) of a shape are differentiable functions of \( t \). Write an equation that relates \( \frac{d A}{d t} \) to \( \frac{d x}{d t} \) \( \frac{d A}{d t}=(\square) \frac{d x}{d t} \)
Ask by Young Dunn. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(\frac{dA}{dt} = 22x \frac{dx}{dt}\)
Solution
Given the area A = 11x², we differentiate both sides with respect to t using the chain rule. This gives:
dA/dt = 11 · 2x · (dx/dt)
Simplifying, we have:
dA/dt = 22x (dx/dt)
So, the box should be filled with "22x".
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Simplify this solution Beyond the Answer
To find the relationship between the rate of change of area \( \frac{d A}{d t} \) and the rate of change of dimension \( \frac{d x}{d t} \), we can use the chain rule. Given \( A = 11 x^2 \), we can differentiate \( A \) with respect to \( t \): \[ \frac{d A}{d t} = \frac{d}{d t}(11 x^2) = 11 \cdot 2x \cdot \frac{d x}{d t} = 22x \cdot \frac{d x}{d t}. \] Thus, we can express this as: \[ \frac{d A}{d t} = 22x \frac{d x}{d t}. \] So, the equation would be \( \frac{d A}{d t} = 22x \frac{d x}{d t} \).
