Suppose a baseball is thrown at 81 miles per hour The ball w

Suppose a baseball is thrown at 81 miles per hour. The ball will travel 346 feet when hit by a bat swung at 54 miles per hour and will travel 470 feet when hit by a bat swung at 85 miles per hour. Let y be the number of feet traveled by the ball when hit by a bat swung at x miles per hour. How much farther will a ball travel for each mile-per-hour increase in the speed of the bat?

Solution

baseball is thrown at 81 miles per hour and hit by a bat swung at 54 miles per hour so effective speed = 81+54 = 135 miles per hour
it travels 346 feet = 346/5280 miles
so point in (x,y) form can be (135,346/5280)

similarly for second part
effective speed = 81+85 = 166
Distance = 470 feet = 470/5280 miles
hence point will be (166,470/5280 )

Now if we assume that there is linear relationship between speed and the distance then we can use equation of line to find the relationship
between speed(x) and distance (y)

slope for points (135,346/5280) and (166,470/5280 ) is given by:
m=(y2-y1)/(x2-x1)=(470/5280 - 346/5280)/(166-135)=1/1320
now plug any point say (166,470/5280 ) and m=1/1320 into point slope formula to find equation
y-y1=m(x-x1)
y-470/5280 =1/1320(x-166)
y-470/5280 =1/1320x-166/1320
y=1/1320x-97/2640

since slope is 1/1320 so that means distance will increase by 1/1320miles or approx 0.000757575757576 miles per increase in speed of the bat in miles per hour.