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Let $d(t)$ be the total number of miles Joanna has cycled, and let $t$ represent the number of hours before stopping for a break during her ride.



$d(t) = 12t + 20$



So, $d(4) =$ [blank]. This means that after [blank], Joanna [blank].

Verified Answer :

  • Substitute \(t = 4\) into the given function \(c(t) = 12t + 20\).
  • Calculate the result: \(c(4) = 12(4) + 20 = 68\).
  • Interpret the result: After 4 hours, Joanna has cycled 68 miles.
  • Final answer placeholders: \(c(4) = \boxed{68}\), after \(\boxed{4}\) hours, Joanna \(\boxed{has cycled 68 miles}\).

Explanation

  1. Analysis of the Given Function

The given function is \(c(t) = 12t + 20\), which describes the total number of miles Joanna has cycled as a function of the number of hours \(t\) she has ridden before her break. We are asked to evaluate the function for \(t = 4\) and interpret the result in the context of the problem. This means substituting \(t=4\) into the expression for \(c(t)\) to determine the distance covered.

  1. Calculating the Function Value

To find the value of \(c(4)\), we substitute \(t = 4\) into the given linear equation:
\[c(4) = 12 \times 4 + 20\]
Performing the multiplication first, we have \(12 \times 4 = 48\).
Next, adding 20 gives:
\[c(4) = 48 + 20 = 68\]
Thus, the value of the function at \(t = 4\) is 68.

  1. Interpretation and Final Answer

In the context of this problem, the variable \(t\) represents the number of hours Joanna has cycled. Therefore, evaluating \(c(4)\) tells us the total distance in miles Joanna has cycled after 4 hours of riding.
Based on our calculation, after 4 hours, Joanna has cycled 68 miles.
Filling in the requested format:

  • \(c(4) = 68\)
  • After 4 hours
  • Joanna has cycled 68 miles.

Examples

Mathematical functions like \(c(t) = 12t + 20\) are common in everyday life for modeling costs or distances. For example, if you are renting a car, the company might charge a base fee of \$\(20\$ plus \$\(12\$ per hour. By using this function, you can quickly determine that for a 4-hour rental, the total cost would be \(12(4) + 20 = \boxed{\$68}\). Modeling real-world scenarios with linear functions allows for quick and accurate decision-making when budgeting time or money.

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