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Chapter 10: Problem 7

$$ R(t)=60 t-2 t^{2} ; t=3 $$

Short Answer

Expert verified
Given the function \(R(t) = 60t - 2t^2\), we need to find the value of R when \(t = 3\). By substituting and simplifying, we obtain \(R(3) = 162\).

Step by step solution

01

Identify the given function and desired input value

The function that we're working with is \(R(t) = 60t - 2t^2\), and we want to find the value of R when \(t = 3\).
02

Substitute t = 3 into the function

To find the value of R when t = 3, we need to substitute t = 3 into the equation: \(R(3) = 60(3) - 2(3)^2\)
03

Simplify the equation

Now, we can simplify the equation. First, let's calculate the values inside the parentheses: \(60(3) = 180\) \((3)^2 = 9\) \(R(3) = 180 - 2(9)\) Next, we need to further simplify the equation: \(R(3) = 180 - 18\)
04

Calculate the final value of R

Finally, we can calculate the value of R when \(t = 3\): \(R(3) = 162\) So, the value of R when \(t = 3\) is 162.

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Key Concepts

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

Function Evaluation
Function evaluation is a crucial part of understanding quadratic functions, specifically when working with equations like \( R(t) = 60t - 2t^2 \). Evaluating a function at a specific value involves determining what the function output is when the input, in this case \( t \), is set to a certain value.
When dealing with the equation \( R(t) = 60t - 2t^2 \), and the task is to find \( R(3) \), we're simply plugging \( 3 \) in place of \( t \). This process helps us understand how the function behaves for that specific value.
  • Choose the specific input, such as \( t = 3 \).
  • Substitute the input into the function.
  • Compute to find the output, which gives us the result of the function at that input.
Function evaluation is like asking a question about the function and finding the precise answer, which is essential in many mathematical problems.
Algebraic Substitution
Algebraic substitution plays a pivotal role when evaluating functions. It's all about replacing variables with specific numbers to make calculations possible.
This is the step where you take the function's original expression and insert the value for which you're evaluating the function into it. Consider our function \( R(t) = 60t - 2t^2 \). If we're evaluating \( R(3) \), we replace every \( t \) in the equation with \( 3 \):
  • Start with the original function: \( R(t) = 60t - 2t^2 \).
  • Substitute \( t = 3 \) into the function.
  • Form a new equation: \( R(3) = 60(3) - 2(3)^2 \).
By substituting, you're simplifying the function so it can be evaluated numerically. It鈥檚 a straightforward yet powerful algebraic technique that transforms the abstract algebra into something tangible.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form to make calculations easier and clearer. After substitution in our example, simplifying \( R(3) = 60(3) - 2(3)^2 \) lays out the final value.
Here's how you simplify step-by-step:
  • Compute multiplication inside the equation: \( 60(3) = 180 \).
  • Calculate the power and subsequent multiplication: \( (3)^2 = 9 \) and then \( 2(9) = 18 \).
  • Subtract the results: \( 180 - 18 = 162 \).
This step-by-step reduction from a complex expression to a single numerical answer exemplifies how simplifying helps decode complex-looking algebra, making the mathematics much more manageable and intuitive. It鈥檚 essential because it keeps work neat and precise, often leading to the final answer of any algebraic problem.

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Most popular questions from this chapter

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=m x+b $$

Suppose the demand for an old brand of \(\mathrm{TV}\) is given by $$ q=\frac{100,000}{p+10} $$ where \(p\) is the price per TV set, in dollars, and \(q\) is the number of TV sets that can be sold at price \(p\). Find \(q(190)\) and estimate \(q^{\prime}(190)\). Interpret your answers. HINT [See Example 1.]

Compute the indicated derivative. $$ U(t)=5.1 t^{2}-1.1 t ; U^{\prime}(1) $$

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Weekly sales of an old brand of TV are given by $$ S(t)=100 e^{-t / 5} $$ sets per week, where \(t\) is the number of weeks after the introduction of a competing brand. Estimate \(S(5)\) and \(\left.\frac{d S}{d t}\right|_{t=5}\) and interpret your answers.
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