Textbook Question
The volume V of a sphere of radius r changes over time t.
b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?
4. Applications of Derivatives
Related Rates
2:51 minutesProblem 12b
Textbook Question
The sides of a square decrease in length at a rate of 1 m/s.
b. At what rate are the lengths of the diagonals of the square changing?
Verified step by step guidance1
Start by understanding the relationship between the side length of the square and its diagonal. If the side length of the square is 's', then the diagonal 'd' can be found using the Pythagorean theorem: \( d = \sqrt{2} \cdot s \).
Differentiate the equation for the diagonal with respect to time 't' to find the rate of change of the diagonal. This gives us \( \frac{dd}{dt} = \sqrt{2} \cdot \frac{ds}{dt} \).
We know from the problem that the side length 's' is decreasing at a rate of 1 m/s, so \( \frac{ds}{dt} = -1 \) m/s.
Substitute \( \frac{ds}{dt} = -1 \) m/s into the differentiated equation: \( \frac{dd}{dt} = \sqrt{2} \cdot (-1) \).
Simplify the expression to find the rate at which the diagonal is changing. This will give you the rate of change of the diagonal in meters per second.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the rate of change of the square's side length affects the rate of change of its diagonal length. This concept is essential for solving problems where multiple variables are interdependent.
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Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For a square, the diagonal can be calculated using this theorem, where the diagonal is the hypotenuse and the sides are the legs of the triangle formed. This relationship is crucial for finding the diagonal's length as the sides change.
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Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of that function with respect to a variable. In this context, we will differentiate the formula for the diagonal length of the square with respect to time to find how fast the diagonal is changing as the side lengths decrease.
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