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Chapter 3: Problem 51

A monthly tablet plan costs 34.99\( . It includes 3 gigabytes of data and charges 15\) per gigabyte for additional gigabytes. The following function is used to compute the monthly cost for a subscriber. \(C(x)=\left\\{\begin{array}{rlr}34.99 & \text { if } 0 \leq x \leq 3 \\ 15 x-10.01 & \text { if } & x>3\end{array}\right.\) Compute the monthly cost for each of the following gigabytes of use. (a) 2 (b) 5 (c) 13

Short Answer

Expert verified
For 2 GB: 34.99. For 5 GB: 64.99. For 13 GB: 184.99.

Step by step solution

01

Understanding the Monthly Cost Function

The cost function is defined piecewise. For 0 to 3 gigabytes, the cost is $34.99. For usage greater than 3 gigabytes, the cost is computed using the formula \(15x - 10.01\), where \(x\) is the number of gigabytes used.
02

Calculate the Monthly Cost for 2 Gigabytes

Using the cost function for \(0 \leq x \leq 3\), the monthly cost for 2 gigabytes is 34.99 since 2 is within the range of 0 to 3 gigabytes.
03

Calculate the Monthly Cost for 5 Gigabytes

Since 5 is greater than 3, use the formula for \(x > 3\): \(15x - 10.01\). Substitute \(x = 5\) into the formula: \[15(5) - 10.01 = 75 - 10.01 = 64.99\]. Thus, the monthly cost for 5 gigabytes is 64.99.
04

Calculate the Monthly Cost for 13 Gigabytes

Again, since 13 is greater than 3, use the formula for \(x > 3\): \(15x - 10.01\). Substitute \(x = 13\) into the formula: \[15(13) - 10.01 = 195 - 10.01 = 184.99\]. Thus, the monthly cost for 13 gigabytes is 184.99.

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Key Concepts

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

Monthly Cost Calculation
Calculating the monthly cost of a tablet plan depends on how much data you use. If your usage is between 0 to 3 gigabytes, you pay a fixed fee of 34.99 dollars. However, once you exceed 3 gigabytes, the cost increases. The additional charge is computed using the formula: \(15x - 10.01\), where \(x\) is the number of gigabytes used.
Let's clarify with an example. If you use 2 gigabytes in a month, you pay the base cost of 34.99 dollars, since your usage is within the 0 to 3 gigabytes range. But if you use 5 gigabytes, you need to calculate the charge for the extra data: \[15(5) - 10.01 = 64.99 \]. Here, you exceed the 3-gigabyte limit, making your total cost 64.99 dollars.
Tablet Plan
A tablet plan typically comes with a fixed data allowance and a base monthly cost. In our scenario, the base cost is 34.99 dollars, which provides 3 gigabytes of data. This structure is common because it offers simplicity—it’s easy for customers to understand and budget for a predictable monthly fee.
  • Fixed Data Allowance: Includes up to 3 gigabytes at a steady rate.
  • Base Monthly Cost: 34.99 dollars for monthly subscription.
The idea is to give users enough data for common tasks like browsing and small downloads. If users need more data, the plan shifts and additional charges apply.
Gigabyte Usage Cost
When you exceed the fixed data allowance in your tablet plan, additional costs come into play. For our example, the usage cost formula kicks in when you use more than 3 gigabytes. The formula \(15x - 10.01\) helps you calculate the extra charge. Here's a step-by-step guide:
First, understand at what point extra charges start. In this case, it's at 3 gigabytes. If you use 5 gigabytes in total, you calculate the cost for the extra 2 gigabytes:
  • \texceeding usage: 5 - 3 = 2 gigabytes
  • \textra charge per gigabyte: 15 * 2 = 30 dollars
  • \tfinal cost: 34.99 (base cost) + 30 - 10.01 = 64.99 dollars
So, understanding this cost structure ensures you can manage your data usage efficiently and anticipate any extra charges.

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Most popular questions from this chapter

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=2 x^{2}-3 x+1\)

\(h(x)=-2 x^{2}+x\) (a) Find the average rate of change from 0 to 3 . (b) Find an equation of the secant line containing \((0, h(0))\) and \((3, h(3))\)

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