Answer :
To determine the average rate at which the object falls during the first 3 seconds of its fall, we want to calculate the average rate of change of the height function [tex]\( h(t) = 300 - 16t^2 \)[/tex].
The average rate of change of a function [tex]\( h(t) \)[/tex] over an interval [tex]\([t_1, t_2]\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1} \][/tex]
In this case, we are looking at the interval from [tex]\( t_1 = 0 \)[/tex] to [tex]\( t_2 = 3 \)[/tex].
1. First, we need to find [tex]\( h(0) \)[/tex].
[tex]\[ h(0) = 300 - 16 \cdot 0^2 = 300 \][/tex]
2. Next, we need to find [tex]\( h(3) \)[/tex].
[tex]\[ h(3) = 300 - 16 \cdot 3^2 = 300 - 16 \cdot 9 = 300 - 144 = 156 \][/tex]
3. Now, we use these values to find the average rate of change over the interval [tex]\( [0, 3] \)[/tex].
[tex]\[ \text{Average Rate of Change} = \frac{h(3) - h(0)}{3 - 0} = \frac{156 - 300}{3} = \frac{-144}{3} = -48 \][/tex]
Therefore, the average rate at which the object falls during the first 3 seconds is:
[tex]\[ \text{Average Rate of Change} = -48 \text{ feet per second} \][/tex]
Out of the given expressions, the correct one that determines the average rate at which the object falls during the first 3 seconds is:
[tex]\[ \frac{h(3) - h(0)}{3} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\frac{h(3) - h(0)}{3}} \][/tex]
The average rate of change of a function [tex]\( h(t) \)[/tex] over an interval [tex]\([t_1, t_2]\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1} \][/tex]
In this case, we are looking at the interval from [tex]\( t_1 = 0 \)[/tex] to [tex]\( t_2 = 3 \)[/tex].
1. First, we need to find [tex]\( h(0) \)[/tex].
[tex]\[ h(0) = 300 - 16 \cdot 0^2 = 300 \][/tex]
2. Next, we need to find [tex]\( h(3) \)[/tex].
[tex]\[ h(3) = 300 - 16 \cdot 3^2 = 300 - 16 \cdot 9 = 300 - 144 = 156 \][/tex]
3. Now, we use these values to find the average rate of change over the interval [tex]\( [0, 3] \)[/tex].
[tex]\[ \text{Average Rate of Change} = \frac{h(3) - h(0)}{3 - 0} = \frac{156 - 300}{3} = \frac{-144}{3} = -48 \][/tex]
Therefore, the average rate at which the object falls during the first 3 seconds is:
[tex]\[ \text{Average Rate of Change} = -48 \text{ feet per second} \][/tex]
Out of the given expressions, the correct one that determines the average rate at which the object falls during the first 3 seconds is:
[tex]\[ \frac{h(3) - h(0)}{3} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\frac{h(3) - h(0)}{3}} \][/tex]
