Question

Problem:
These are the fees for sending a package by Media Mail:

$\bullet$ $\$2.41$ for the first pound

$\bullet$ $\$0.41$ per additional pound for the next six pounds

$\bullet$ $\$0.39$ per additional pound after that

For example, a two-pound package costs $\$2.82$ to send by Media Mail.

If a package full of books costs $\$7.99$ to send by Media Mail, how many pounds does the package weigh? Explain your work in complete sentences.

Answer 1

Answer:

15 pounds

Step-by-step explanation:

Let's denote the weight of the package in pounds as $\bold{\sf x}$.

The cost of sending the package is composed of three parts:

• $\sf \ 2.41$ for the first pound,
• $\sf \ 0.41$ per additional pound for the next six pounds, and
• $\sf \ 0.39$ per additional pound after that.

The cost function $\bold{\sf C(x)}$ for sending a package of weight $\bold{\sf x}$ is given by:

$\sf C(x) = 2.41 + 0.41(6) + 0.39(x - 7)$

This expression represents the cost of the first pound, the cost of the next six pounds, and the cost of any additional pounds beyond the first seven.

Now, let's set up an equation based on the given information:

$\sf C(x) = 7.99$

Substitute the expression for $\bold{\sf C(x)}$ into the equation:

$\sf 2.41 + 0.41(6) + 0.39(x - 7) = 7.99$

Now, solve for $\bold{\sf x}$:

$\sf 2.41 + 2.46 + 0.39(x - 7) = 7.99$

Combine like terms:

$\sf 4.87 + 0.39(x - 7) = 7.99$

Subtract 4.87 from both sides:

$\sf 4.87 + 0.39(x - 7) -4.87 = 7.99-4.87$

$\sf 0.39(x - 7) = 3.12$

Divide by 0.39:

$\sf x - 7 = \dfrac{3.12}{0.39}$

$\sf x = \dfrac{3.12}{0.39} + 7$

$\sf x \approx 15$

Therefore, the package weighs approximately 15 pounds.

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