A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 19 ft/s.

(a) At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.)

ft/s

(b) At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)

ft/s

Draw a diagram. It is clear that when the batter has gone x feet, his distance z from 2nd base is

z^2 = (90-x)^2 + 90^2
so,
2z dz/dt = 2(x-90) dx/dt
so, figure z when x=45 and plug in the numbers

The distance from 3rd base is very similar.

To solve this problem, we will use the concept of related rates. Let's start by breaking down the given information:

- The baseball diamond is a square with side length 90 ft.
- The batter is running toward first base with a speed of 19 ft/s.

Let's label the corners of the square diamond as the bases: first base (FB), second base (SB), third base (TB), and home plate (HP).

(a) To find the rate at which the batter's distance from second base is decreasing when he is halfway to first base, we need to find the distance between the batter and second base when he is halfway to first base. Let's call this distance "x".

Using the Pythagorean theorem, we know that the distance between two points in a square is given by the formula:

d^2 = x^2 + (side length)^2

Substituting the given values, we get:

x^2 = (90 ft)^2 + (90 ft)^2
x^2 = 8100 ft^2 + 8100 ft^2
x^2 = 16200 ft^2

Taking the square root of both sides, we find:

x = √16200 ft
x ≈ 127.3 ft

Now we need to find the rate at which x is changing. We are given that the batter's speed is 19 ft/s, so dx/dt = 19 ft/s.

Using the chain rule, we can find the rate at which x is changing with respect to time:

dx/dt = ds/dt - ds/dt, where s is the distance between the batter and second base

Since the distance between the batter and second base is changing at a constant rate (the batter is running in a straight line towards first base), ds/dt is equal to the batter's speed:

ds/dt = 19 ft/s

Therefore, the rate at which the batter's distance from second base is decreasing when he is halfway to first base is equal to dx/dt - ds/dt:

dx/dt - ds/dt = 19 ft/s - 19 ft/s
= 0 ft/s

So, the rate at which the batter's distance from second base is decreasing when he is halfway to first base is 0 ft/s.

(b) To find the rate at which the batter's distance from third base is increasing at the same moment, we need to find the distance between the batter and third base when he is halfway to first base. Let's call this distance "y".

Using the Pythagorean theorem, we have:

y^2 = (90 ft)^2 + (90 ft)^2
y^2 = 16200 ft^2

Taking the square root of both sides, we find:

y = √16200 ft
y ≈ 127.3 ft

Now we need to find the rate at which y is changing. Again, we are given that dx/dt = 19 ft/s.

Using the chain rule, we can find the rate at which y is changing with respect to time:

dy/dt = ds/dt + ds/dt

Since the distance between the batter and third base is changing at a constant rate, ds/dt is equal to the batter's speed:

ds/dt = 19 ft/s

Therefore, the rate at which the batter's distance from third base is increasing at the same moment is equal to dy/dt + ds/dt:

dy/dt + ds/dt = 19 ft/s + 19 ft/s
= 38 ft/s

So, the rate at which the batter's distance from third base is increasing at the same moment is 38 ft/s.

To solve this problem, we need to use the concept of related rates. Related rates involve finding how the rates of change of two or more variables are related to each other.

Let's break down the problem step by step:

(a) At what rate is his distance from second base decreasing when he is halfway to first base?

Let's call the distance from the batter to second base "x" and the distance from the batter to first base "y." We want to find the rate at which x is decreasing when y is equal to half its length.

Given:
- Side length of the baseball diamond (which forms a square) = 90 ft
- Speed of the batter towards first base (dy/dt) = 19 ft/s
- Find the rate at which x (distance from batter to second base) is decreasing when y (distance from batter to first base) is half of its length.

To solve this, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides:

x^2 + y^2 = 90^2

Differentiate both sides of the equation with respect to time (t):

2x(dx/dt) + 2y(dy/dt) = 0

Since we are interested in finding dx/dt when y = 45 (half its length), we substitute the known values into the equation:

2x*dx/dt + 2(45)(19) = 0

Simplifying the equation:

2x*dx/dt = -2(45)(19)

Divide both sides by 2x to isolate dx/dt:

dx/dt = -2(45)(19) / (2x)

We know that x = 90 - y, where y is the distance to first base, so we substitute this into the equation:

dx/dt = -2(45)(19) / (2(90 - y))

At halfway to first base, y = 45, so we can substitute it into the equation:

dx/dt = -2(45)(19) / (2(90 - 45))

Simplifying further:

dx/dt = -2(45)(19) / (2(45))

Cancelling out the common factors:

dx/dt = -19 ft/s

So, the rate at which the distance from the batter to second base is decreasing when he is halfway to first base is -19 ft/s.

(b) At what rate is his distance from third base increasing at the same moment?

Since the baseball diamond is a square, the distance from the batter to third base is also 90 ft. Therefore, the distance from the batter to third base is a constant and not changing.

Hence, the rate at which his distance from third base is increasing is 0 ft/s.