Abaseball diamond is a square with side 90 ft. a batter hits the ball and runs toward first base with a speed of 26 ft/s.
(a) at what rate is his distance from second base decreasing when he is halfway to first base?
ft/s
(b) at what rate is his distance from third base increasing at the same moment?
ft/s

(a) Distance decreases at a rate of 11.628 ft/s.

(b) Distance increases at a rate of 11.628 ft/s.

Step-by-step explanation:

Refer to the diagram attached to at the bottom of this answer. We will call a the distance between the batter and 1st base, x the distance between the batter and 2nd base and y the distance between the batter and 3rd base.

The speed of the batter toward first base is 26 ft/s, which means:

Since the diamond is a square, the angle between trajectories in each base is a square angle, so we can use Pythagoras theorem:

Finding the first derivative:

When the batter is halfway to 1st, a = 45 ft, then:

So the distance between the batter and 2nd base decreases at 11.628 ft/s

Now, using Pythagoras again to find y:

and we know that

Finding the first derivative:

at  b = 45 ft

So the distance between the batter and 3rd base increases at 11.628 ft/s

since your problem is x^4 = 81, what you have to undeo is the "to the fourth power" part.

the inverse (opposite operation) of "to the fourth power" is "to the fourth root".

  (x^4) ^ ¼   = 81 ^ ¼  

  81 = 3^4     so (3 ^ 4) ^ ¼   =   3 ^ 1   = 3

by hand: (3 ^ 4) ^ ¼   =   3 ^ (4 x ¼) = 3 ^ 1   = 3   (when you take a power to a power, multiply the exponents.)

or, do it by calculator. one way of finding the fourth root is to use ^ (1/4)

step-by-step explanation: