Q. 53 - Calculus

Question

Suppose h(t) represents the average height, in feet, of a person who is t years old. 

(a)  In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units? 

(b)  Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why? 

(c)  At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why? 

Step-by-Step Solution

Answer

Ans:  

(a)   Here $h\left(12\right)$) represents the height of the person when he was $12$ years old.

And $h\text{'}\left(12\right)$ represents the average rate of change of the height of the person when he was $12$ years old.

(b)    Here $h\left(12\right)$ represent the height of the person when he was $12$ years old. Therefore it is a positive quantity.

And $h\text{'}\left(12\right)$ represents the average rate of change of the height of the person when he was $12$ years old. Since the height of a person increases with his age, therefore it is also a positive quantity.

(c)   Here $h\left(t\right)$ represents the height of the person when he was $t$ years old. Therefore the maximum height of the person will be attained between the years $20-25$.

Again $h\text{'}\left(t\right)$ represents the average rate of change of the height of the person after $t$ years. It is maximum when a person is $12-15$ years old.

Step 1. Given information.

given, 

         $h\left(t\right)$ represents the average height in feet.

         where $t$ represents the years the person is old.

Step 2. (a) Here the objective is to write the unit of h ( 12 ) and h ' ( 12 ) .

Here $h\left(12\right)$ represents the height of the person when he was 12 years old.

Again $h\text{'}\left(12\right)$ represents the average rate of change of the height of the person when he was $12$ years old.

Step 3. (b) Here the objective is to write whether h ( 12 ) and h ' ( 12 ) are positive or negative.

Here $h\left(12\right)$ represents the height of the person when he was $12$ years old. Therefore it is a positive quantity.

Again $h\text{'}\left(12\right)$ represents the average rate of change of the height of the person when he was $12$ years old. Since the height of a person increases with his age, therefore it is also a positive quantity.

Step 4. (c) Here the objective is to write when h ( t ) and h ' ( t ) is maximum.

Here $h\left(t\right)$ represents the height of the person when he was t years old. Therefore the maximum height of the person will be attained between the years $20-25$.

Again $h\text{'}\left(t\right)$ represent the average rate of change of the height of the person after $t$ years. It is maximum when a person is $12-15$ years old.