Math / Algebra

Question2. The cost, $C$ , in dollars, of shipping a package that weighs $w$ pounds can be modeled by the function given by $C(w)=\left\{\begin{array}{ll} 11 & 0<w<2 \\ 9+1.49 w & 2 \leq w<10 \\ 6+1.99 w & w \geq 10 \end{array}\right.$ a. How much does it cost to ship a package that weighs 1.5 pounds? b. How much does it cost to ship a package that weighs 5 pounds? c. How much more does it cost to ship a package that weighs 10 pounds than a package that weighs 9 pounds?

Studdy Solution

STEP 1

What is this asking? We're given a funky function that tells us the cost of shipping a package based on its weight, and we need to figure out the cost for a few different weights and compare them. Watch out! Make sure to use the correct part of the function depending on the weight!
Don't mix them up!

STEP 2

1. Cost for 1.5 pounds
2. Cost for 5 pounds
3. Cost difference between 10 and 9 pounds

STEP 3

Alright, so we're shipping a package that weighs $1.5$ pounds.
That's between $0$ and $2$ , so we're gonna use the first part of our cost function: $C(w) = 11$ .
This means no matter how much the package weighs (as long as it's between $0$ and $2$ pounds), it costs $\$11$ to ship.

STEP 4

Since our package weighs $1.5$ pounds, and $0 < 1.5 < 2$ , we plug in $w = 1.5$ into the first part of the function.
Even though we don't see a $w$ in that part of the function, it's still there, hidden!
It's like a secret agent.
It's saying the cost is always $11$ when the weight is between $0$ and $2$ .
So, $C(1.5) = 11$ .

STEP 5

Now, we've got a 5-pound package.
This time, we look at our function, and $5$ falls between $2$ and $10$ , so we use the second part of the function: $C(w) = 9 + 1.49w$ .

STEP 6

Let's plug in $w = 5$ : $C(5) = 9 + 1.49 \cdot 5$ .

STEP 7

Calculating that out, $1.49 \cdot 5 = 7.45$ , so $C(5) = 9 + 7.45 = 16.45$ .
So, it costs $\$16.45$ to ship a 5-pound package.

STEP 8

For a 10-pound package, we use the third part of the function: $C(w) = 6 + 1.99w$ , because $10$ is greater than or equal to $10$ .
Plugging in $w = 10$ , we get $C(10) = 6 + 1.99 \cdot 10 = 6 + 19.90 = 25.90$ .

STEP 9

For a 9-pound package, we use the second part of the function: $C(w) = 9 + 1.49w$ , because $9$ is between $2$ and $10$ .
Plugging in $w = 9$ , we get $C(9) = 9 + 1.49 \cdot 9 = 9 + 13.41 = 22.41$ .

STEP 10

To find how much more it costs to ship the 10-pound package, we subtract the cost of the 9-pound package from the cost of the 10-pound package: $25.90 - 22.41 = 3.49$ .

STEP 11

a. It costs $\$11$ to ship a $1.5$ -pound package. b. It costs $\$16.45$ to ship a $5$ -pound package. c. It costs $\$3.49$ more to ship a $10$ -pound package than a $9$ -pound package.

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Studdy Solution

STEP 1

STEP 2

1. Cost for 1.5 pounds
2. Cost for 5 pounds
3. Cost difference between 10 and 9 pounds

STEP 3

Alright, so we're shipping a package that weighs $1.5$ pounds.
That's between $0$ and $2$ , so we're gonna use the first part of our cost function: $C(w) = 11$ .
This means no matter how much the package weighs (as long as it's between $0$ and $2$ pounds), it costs $\$11$ to ship.

STEP 4

STEP 5

Now, we've got a 5-pound package.
This time, we look at our function, and $5$ falls between $2$ and $10$ , so we use the second part of the function: $C(w) = 9 + 1.49w$ .

STEP 6

Let's plug in $w = 5$ : $C(5) = 9 + 1.49 \cdot 5$ .

STEP 7

Calculating that out, $1.49 \cdot 5 = 7.45$ , so $C(5) = 9 + 7.45 = 16.45$ .
So, it costs $\$16.45$ to ship a 5-pound package.

STEP 8

For a 10-pound package, we use the third part of the function: $C(w) = 6 + 1.99w$ , because $10$ is greater than or equal to $10$ .
Plugging in $w = 10$ , we get $C(10) = 6 + 1.99 \cdot 10 = 6 + 19.90 = 25.90$ .

STEP 9

For a 9-pound package, we use the second part of the function: $C(w) = 9 + 1.49w$ , because $9$ is between $2$ and $10$ .
Plugging in $w = 9$ , we get $C(9) = 9 + 1.49 \cdot 9 = 9 + 13.41 = 22.41$ .

STEP 10

To find how much more it costs to ship the 10-pound package, we subtract the cost of the 9-pound package from the cost of the 10-pound package: $25.90 - 22.41 = 3.49$ .

STEP 11

a. It costs $\$11$ to ship a $1.5$ -pound package. b. It costs $\$16.45$ to ship a $5$ -pound package. c. It costs $\$3.49$ more to ship a $10$ -pound package than a $9$ -pound package.

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Studdy Solution

STEP 1

STEP 2

1. Cost for 1.5 pounds2. Cost for 5 pounds3. Cost difference between 10 and 9 pounds

STEP 3

Alright, so we're shipping a package that weighs 1.51.51.5 pounds.That's between 000 and 222, so we're gonna use the first part of our cost function: C(w)=11C(w) = 11C(w)=11.This means no matter how much the package weighs (as long as it's between 000 and 222 pounds), it costs $11\$11$11 to ship.

STEP 4

STEP 5

Now, we've got a **5-pound package**.This time, we look at our function, and 555 falls between 222 and 101010, so we use the second part of the function: C(w)=9+1.49wC(w) = 9 + 1.49wC(w)=9+1.49w.

STEP 6

Let's plug in w=5w = 5w=5: C(5)=9+1.49⋅5C(5) = 9 + 1.49 \cdot 5C(5)=9+1.49⋅5.

STEP 7

Calculating that out, 1.49⋅5=7.451.49 \cdot 5 = 7.451.49⋅5=7.45, so C(5)=9+7.45=16.45C(5) = 9 + 7.45 = 16.45C(5)=9+7.45=16.45.So, it costs $16.45\$16.45$16.45 to ship a **5-pound package**.

STEP 8

STEP 9

For a **9-pound package**, we use the second part of the function: C(w)=9+1.49wC(w) = 9 + 1.49wC(w)=9+1.49w, because 999 is between 222 and 101010.Plugging in w=9w = 9w=9, we get C(9)=9+1.49⋅9=9+13.41=22.41C(9) = 9 + 1.49 \cdot 9 = 9 + 13.41 = 22.41C(9)=9+1.49⋅9=9+13.41=22.41.

STEP 10

To find how much *more* it costs to ship the **10-pound package**, we subtract the cost of the **9-pound package** from the cost of the **10-pound package**: 25.90−22.41=3.4925.90 - 22.41 = 3.4925.90−22.41=3.49.

STEP 11

a. It costs $11\$11$11 to ship a 1.51.51.5-pound package. b. It costs $16.45\$16.45$16.45 to ship a 555-pound package. c. It costs $3.49\$3.49$3.49 more to ship a 101010-pound package than a 999-pound package.

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1. Cost for 1.5 pounds
2. Cost for 5 pounds
3. Cost difference between 10 and 9 pounds

Alright, so we're shipping a package that weighs $1.5$ pounds.
That's between $0$ and $2$ , so we're gonna use the first part of our cost function: $C(w) = 11$ .
This means no matter how much the package weighs (as long as it's between $0$ and $2$ pounds), it costs $\$11$ to ship.

Now, we've got a 5-pound package.
This time, we look at our function, and $5$ falls between $2$ and $10$ , so we use the second part of the function: $C(w) = 9 + 1.49w$ .

Let's plug in $w = 5$ : $C(5) = 9 + 1.49 \cdot 5$ .

Calculating that out, $1.49 \cdot 5 = 7.45$ , so $C(5) = 9 + 7.45 = 16.45$ .
So, it costs $\$16.45$ to ship a 5-pound package.

For a 9-pound package, we use the second part of the function: $C(w) = 9 + 1.49w$ , because $9$ is between $2$ and $10$ .
Plugging in $w = 9$ , we get $C(9) = 9 + 1.49 \cdot 9 = 9 + 13.41 = 22.41$ .

To find how much more it costs to ship the 10-pound package, we subtract the cost of the 9-pound package from the cost of the 10-pound package: $25.90 - 22.41 = 3.49$ .

a. It costs $\$11$ to ship a $1.5$ -pound package. b. It costs $\$16.45$ to ship a $5$ -pound package. c. It costs $\$3.49$ more to ship a $10$ -pound package than a $9$ -pound package.