A baseball diamond is a square with sides of length 90ft. Suppose Manny hits a "double", so that he must run from home plate to second base. He runs at a constant speed of 15 feet per second. Express the distance (in feet) between Manny and home plate as a function of time t (seconds).
Answer
f(x)={??? if 0 ≤ t ≤ 6
{??? if 6 ≤ t ≤ 12
To determine the distance between Manny and home plate as a function of time, we need to consider Manny's position on the baseball diamond during different time intervals.
Since Manny hits a "double," he starts at home plate and runs to second base. The baseball diamond is a square with sides of length 90 feet, so the distance from home plate to second base is also 90 feet.
Let's break down the time intervals and calculate the distances accordingly:
1. For the interval 0 ≤ t ≤ 6 seconds:
During this time, Manny is running from home plate towards first base. As he runs at a constant speed of 15 feet per second, the distance he covers can be calculated using the formula: distance = speed × time.
Since Manny runs at 15 feet per second for 6 seconds, the distance he covers is:
distance = 15 feet/second × 6 seconds = 90 feet.
Therefore, the distance between Manny and home plate during this interval is 90 feet.
2. For the interval 6 ≤ t ≤ 12 seconds:
During this time, Manny is running from first base towards second base. As he continues to run at a constant speed of 15 feet per second, the distance he covers can be calculated in the same manner:
distance = 15 feet/second × (t - 6)
= 15t - 90 feet.
Therefore, the distance between Manny and home plate during this interval (6 ≤ t ≤ 12) can be expressed as 15t - 90 feet.
Putting it all together, the function that expresses the distance between Manny and home plate as a function of time t is:
f(t) = 90 feet for 0 ≤ t ≤ 6 seconds
f(t) = 15t - 90 feet for 6 ≤ t ≤ 12 seconds.
To find the distance between Manny and home plate as a function of time, we need to determine the distance he has covered during different time intervals.
First, we need to determine the total distance Manny needs to run, which is the diagonal of the square baseball diamond. The diagonal of a square can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the diagonal represents the distance between home plate and second base, and the sides of the square represent the distance between any two consecutive bases.
Using the Pythagorean theorem, we can calculate the diagonal distance:
diagonal = √(90^2 + 90^2)
= √(8100 + 8100)
= √16200
= 127.28 feet (approximately)
Since Manny is running at a constant speed of 15 feet per second, we can break down the total time taken into intervals of 6 seconds each (since Manny's speed is 15 feet/s, and the distance covered during 6 seconds would be 15 * 6 = 90 feet, which is the length of each side of the square baseball diamond).
Now, let's break down the time intervals and determine the distance covered during each interval:
1. For 0 ≤ t ≤ 6 seconds:
Manny has covered a distance equal to his constant speed multiplied by the time elapsed.
distance = 15 * t
2. For 6 ≤ t ≤ 12 seconds:
Manny has reached the second base (which is 90 feet away from home plate),
so the distance remains constant at 90 feet.
Therefore, the function representing the distance between Manny and home plate as a function of time is:
f(t) = 15t if 0 ≤ t ≤ 6
90 if 6 ≤ t ≤ 12
Draw a diagram!
clearly, if f(t) is the distance in feet,
for 0<=t<=6, f(t) = 15t
for 6<=t<=12, f(t) = √(90^2+(15t)^2)