Bases are located on the field 90 feet away from one another Jimmy is running at a speed of 10 ft/sec from second to third base. When Jimmy is halfway to third base, how quickly is the distance between him and home plate decreasing?

Question

William Abdalla · Accepted Answer

Here is a link to a very similar question and its solution.

Use the same picture, but relabel, so that B is second base, A is third and C is home plate. Also note the difference in speeds of the runners. https://tutor.hix.ai

Emma Aldridge · Answer

undefined We can use the Pythagorean theorem to find the distance between Jimmy and home plate at any given point. Let x be the distance from Jimmy to the second base. Then, according to the Pythagorean theorem:

$x^2 + 90^2 = (90 + 10t)^2$

Differentiating both sides with respect to time t, we get:

$2x \frac{dx}{dt} = 2(90 + 10t) \cdot 10$

At the halfway point to third base, $x = 45$ feet. Substituting this value into the equation and solving for $\frac{dx}{dt}$, we find:

$\frac{dx}{dt} = \frac{(90 + 10t) \cdot 10}{x}$

Plugging in $x = 45$ and $t = 90/10 = 9$ (since Jimmy takes 9 seconds to run from second to third base), we get:

$\frac{dx}{dt} = \frac{(90 + 10(9)) \cdot 10}{45} = \frac{180}{3} = 60 \text{ ft/sec}$