Problem 72 Estimate the cost of the next it... [FREE SOLUTION]

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Calculus Applications and Technology

Edmond C. Tomastik

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 4: Problem 72

Estimate the cost of the next item or the revenue or profits from the sale of the next item for the indicated functions and the indicated $x$ $$ P(x)=\sqrt[3]{x}(x-54)-50, x=27 $$

Short Answer

Expert verified

The estimated profit is -131.

Step by step solution

Understand the Problem

We are given a function $P(x) = \sqrt[3]{x}(x-54) - 50$ and we need to estimate the profit from the sale of the next item for $x = 27$. This means we need to calculate $P(27)$.

Substitute the Value of x

Substitute $x = 27$ into the function: \[P(27) = \sqrt[3]{27}(27 - 54) - 50\]

Calculate the Cube Root

First, calculate $\sqrt[3]{27}$. Since 3 cubed is 27, we have:\[\sqrt[3]{27} = 3\]

Multiply with the Subtracted Value

Now, subtract 54 from 27:\[27 - 54 = -27\]Multiply the result by the cube root calculated in the previous step:\[3 \times (-27) = -81\]

Final Calculation

Subtract 50 from the value calculated in Step 4:\[-81 - 50 = -131\]

Conclusion

The estimated profit for the sale of the next item, using the function for $x = 27$, is $-131$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Profit Estimation

Profit estimation in calculus often involves substituting a specific value, usually denoted as $x$, into a profit function $P(x)$. The solution to this function tells us the expected profit, or sometimes loss, from a particular production or sales activity. In the provided exercise, the function includes a cube root calculation and subtraction operation, which affect the profit outcome.

To estimate profits, carefully examine the structure of the profit function. Sometimes certain features (like cube roots or other operations) can dramatically affect profits.
Remember, negative profits indicate a loss, which is crucial for decision-making in business scenarios.

For $x = 27$, when calculated via the given function, the profit yielded is $-131$, indicating a loss of 131 units. Understanding these estimations helps in adjusting future strategies or pricing to avoid losses.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to find its corresponding output. This is a crucial concept in calculus and many other fields where functions are employed to model real-world phenomena. The goal is to see what the function yields when the input $x$ is set to a specific value.

Begin by writing down the function clearly, as was done with $P(x) = \sqrt[3]{x}(x-54) - 50$.
Identify the value of $x$ that needs to be substituted, for instance, $x = 27$.
Substitute $27$ into the function, carefully performing each operation according to the standard order of operations (PEMDAS/BODMAS).

Function evaluation not only provides insight into the behavior of the function at particular points but can also clarify whether the outcomes will be profit or loss related. In practical applications, such an evaluation helps identify turning points or threshold values critical for financial planning.

Cube Root Calculation

The cube root calculation is a fundamental arithmetic operation required to solve many problems, often encountered while evaluating functions. The cube root of a number $a$ is a number $b$ such that $b^3 = a$. In simpler terms, it's the number that, when multiplied by itself twice, results in the initial number.

To calculate the cube root of $27$, recognize that $27 = 3 \times 3 \times 3$. Hence, $\sqrt[3]{27} = 3$.
Unlike square roots, cube roots can yield negative results when dealing with negative values because cubing a negative number results in a negative number, as seen with $(-3)^3 = -27$.

Calculating cube roots might at first seem challenging, but recognizing the cube of common numbers can expedite the process. In calculus and algebra, being proficient in these calculations enhances the ability to unravel complex functions swiftly, as encountered in profit estimations and other similar applications.

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91影视

Estimate the cost of the next item or the revenue or profits from the sale of the next item for the indicated functions and the indicated \(x\) $$ P(x)=\sqrt[3]{x}(x-54)-50, x=27 $$

Short Answer

Step by step solution

Understand the Problem

Substitute the Value of x

Calculate the Cube Root

Multiply with the Subtracted Value

Final Calculation

Conclusion

Key Concepts

Profit Estimation

Function Evaluation

Cube Root Calculation

One App. One Place for Learning.

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