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?

Short 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-by-step solution

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.