#### To determine

**To find:** The rate at which his distance from the second base decreasing when he is halfway to the first base.

#### Answer

The decreasing rate of his baseball distance from the second base is dydt=−245 ft/s.

#### Explanation

**Given:**

Let given that a baseball diamond is a square with side 90 ft.

A batter hits the ball and the ball run towards first base with a speed of 24 ft/s, that is dxdt=24 ft/s .

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Pythagorean Theorem.

**Calculation:**

Let x be the distance of the batter from the base point H, and y be the distance between the batter and the point from baseball at which it dropped.

Since both x and y changes with the time t

Therefore, x and y both are function of the time variable t

The position of the batter and the baseball as shown in the figure-1 given below

Apply Pythagoras Theorem in the right side Right angle triangle,

y2=(90−x)2+902

Obtain the value of y when x=45,

y2=(90−45)2+902=452+902=2,025+8,100=10,125

On further simplification the value of y is as follows.

y=10,125 =45×45×5=455 ft .

Obtain dydt when he is halfway to first base.

Differentiate y with respect to the time t,

ddy[y2]=ddt[(90−x)2+902]2ydydt =2(90−x)ddx(90−x)dxdt2ydydt =2(90−x)(−1)dxdt2ydydt =−2(90−x)

On further simplification the value of dydt is as follows

dydt=−(90−x)y⋅dxdt

Substitutes x=45 and y=455 and dxdt=24 in dydt

dydt=−(90−45)455(24)=−45455(24)=−245 ft/s

Therefore, the decreasing rate of his baseball distance from the second base is dydt=−245 ft/s.

#### To determine

**To find:** The rate at which his distance from the third base increasing when he is halfway to the first base.

#### Answer

The increasing rate of his baseball distance from the third base is dzdt=245 ft/s

#### Explanation

**Formula used:**

Chain rule: dydx=dydu⋅dudx

**Calculation:**

Let x be the distance of the batter from the base point H, and z is the distance between the third base and the batter on the first base is shown in the Figure 1.

Apply Pythagoras Theorem in the left side Right angle triangle in figure-1

z2=x2+902

The value of z when x=45

z2=452+902=2,025+8,100=10,125=10,125

On further simplification the distance between the batter and the third base as follows.

z=45×45×5=455 ft

Differentiate z with respect to the time t.

ddt[z2]=ddt[x2+902]2zdzdt =ddx[x2]2zdzdt =2xdxdtdzdt =xz⋅dxdt

Substitute x=45 and y=455 and dxdt=24 in dzdt

dzdt=(45455)(24)=245 ft/s

Therefore, the increasing rate of his baseball distance from the third base is dzdt=245 ft/s.