For the following exercises, refer to the figure of baseball...

Name: Problem 43, Chapter 4: Calculus Volume 1
Uploaded: 2020-03-19T07:03:03-08:00
Duration: 9 min 35 s
Channel: Bobby Barnes
Description: For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft. A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft. A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.

A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

Key Concepts

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to other quantities whose rates of change are known. This concept is a fundamental application of differentiation in calculus where multiple variables, each a function of time, interact through an equation that can be differentiated implicitly with respect to time.

Law of Cosines

The Law of Cosines is a trigonometric identity that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful in non-right triangles, allowing one to determine an unknown side or angle when the other sides and an enclosed angle are known. In problems involving distances between moving objects, it provides a way to connect different distances through the geometry of the situation.

Chain Rule

The Chain Rule is a basic technique in calculus used for differentiating composite functions. In the context of related rates, many variables depend on time indirectly through other variables, so the chain rule is applied to differentiate these dependencies properly. This ensures that the rate of change of a quantity with respect to time is accurately calculated when it is a function of another variable that also varies with time.

Transcript

00:01 So a batter hits a ball towards second base at a rate of 80 feet per second and run towards the first base at a rate of 30 feet per second.

00:10 We want to determine the rate at which the distance between the ball and the batter change after the runner has covered one third of the distance to the first base.

00:20 And they give us the hint to use the law of cosine.

00:23 So i went ahead and wrote the law of cosines on the board right there.

00:27 Now let's think a little bit more about how that may apply to the law.

00:32 What we're doing.

00:33 So let's say he hits the ball to like here and then the ball is going to be about in that location.

00:46 So actually let's make that a little bit bigger.

00:49 So it looks something kind of like this here.

00:54 So we'll call this here angle a and what we're going to want to find now is what is is this distance, how is it changing? so we want to find da by dt, and then it doesn't really matter what we call these other size b and c.

01:20 But we know the rate of change for b, so d, b by dt, well, if we look at the triangle we have here, that would be like he was running the first base, so that will be 30 feet per second.

01:35 And then for c, this is going to be the rate for the ball going straight to second base.

01:44 So this is going to say that dc by dt, this should be 80 feet per second.

01:54 All right.

01:55 So let's go ahead and take some derivatives of this first, and then we can kind of go from there, because we know we want to find what da by dt is.

02:03 And then once we get there, we can figure out what other things are important that we still need to solve for.

02:09 So taking the derivative of this whole thing with respect to time.

02:13 This is going to give us, so 2a times da by d t, plus, or not plus is equal to, is equal to 2b times db by d t, minus.

02:33 Now our angle in this case, a is going to be constant, because think about it if we keep on moving out like this, the angle between these two lines should stay the same.

02:44 So we can factor that out when we take the derivative, and then it's just like we're taking the derivative of b -time -c, which would be product rule.

02:55 So we'd have b -t times b -c by d -t, and then plus c -t times d -b -d -t.

03:05 All right, so we have that now.

03:07 So it looks like we still need to figure out what our angle a is and what b -a -a -n.

03:16 And c r...

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