Q. 54 - Calculus

Question

A tomato plant given $x$ ounces of fertilizer will successfully bear $T\left(x\right)$ pounds of tomatoes in a growing season.

(a)  In real-world terms, what does $T\left(5\right)$ represent and what are its units? What does $T\text{'}\left(5\right)$ represent and what are its units?

(b)  A study has shown that this fertilizer encourages tomato production when less than $20$ ounces are used, but inhibits production when more than $20$ ounces are used. When is $T\left(x\right)$ positive and when is $T\left(x\right)$ negative? When is $T\text{'}\left(x\right)$ positive and when is $T\text{'}\left(x\right)$ negative? 

Step-by-Step Solution

Answer

Ans:    

(a)   Here $T\left(5\right)$ represents the weight of the tomatoes when $5$ ounces of fertilizer have been used.

And $T\text{'}\left(5\right)$ represents the average rate of change of weight of the tomatoes when $5$ ounces of fertilizer has been used.

(b)   Since the use of less than $20$ ounces increases the weight of the tomatoes, therefore $T\left(x\right)$ is positive and form more than $20$ ounces decreases the weight of the tomatoes, therefore $T\left(x\right)$ is negative.

Again less than $20$ ounces increases the weight of the tomatoes, therefore $T\text{'}\left(x\right)$ is positive, and form more than $20$ ounces decreases the weight of the tomatoes, therefore $T\text{'}\left(x\right)$ is negative.

Step 1. Given information.

given,   $T\left(x\right)$

          where T represents the weight of tomatoes when $x$ ounces of fertilizer is used.

Step 2. Here the objective is to write the unit of T ( 5 ) and T ' ( 5 ) , what they mean in the real world.

Here $T\left(5\right)$ represents the weight of the tomatoes when $5$ ounces of fertilizer has been used.

And $T\text{'}\left(5\right)$ represents the average rate of change of weight of the tomatoes when $5$ ounces of fertilizer has been used.

Step 3. Here the objective is to say when T ( x ) is positive or negative and T ' ( x ) is positive or negative.

It is given that when fertilizer is used less than $20$ ounces then the weight of the tomatoes is increasing and at the same time if more than $20$ ounces has been used then weight becomes less.

Since the use of less than $20$ ounces increases the weight of the tomatoes, therefore $T\left(x\right)$ is positive, and form more than $20$ ounces decreases the weight of the tomatoes, therefore $T\left(x\right)$ is negative.

And  less than $20$ ounces increases the weight of the tomatoes, therefore $T\text{'}\left(x\right)$ is positive and form more than $20$ounces decreases the weight of the tomatoes, therefore $T\text{'}\left(x\right)$ is negative.