Math / Algebra

QuestionThe United States Postal Service charges $\$ 0.55$ to mail a first-class letter that weighs up to 1 . ounce. Each additional ounce costs $\$ 0.15$ . Which equation represents the situation? $y=0.15\lfloor x-1\rfloor 0.5$ $y=0.55\lfloor x-1\rfloor+0.15$ $y=0.55\lfloor x\rfloor+0.15$ $y=0.15\lfloor x\rfloor+0.55$

Studdy Solution

STEP 1

What is this asking? We need to find an equation that tells us the cost of mailing a letter based on its weight. Watch out! The first ounce has a different price than the rest, so be careful!

STEP 2

1. Define variables
2. Build the equation
3. Test the equation

STEP 3

Let's use $y$ to represent the total cost in $\$$ of mailing the letter.

STEP 4

Let's use $x$ to represent the weight of the letter in ounces.

STEP 5

The first ounce costs $\$0.55$ .
This is our starting point.

STEP 6

Each additional ounce after the first ounce costs $\$0.15$ .

STEP 7

We can use the greatest integer function (also known as the floor function), written as $\lfloor x \rfloor$ , which gives us the largest integer less than or equal to $x$ .
For example, $\lfloor 2.3 \rfloor = 2$ and $\lfloor 1 \rfloor = 1$ .

STEP 8

For every ounce after the first, we add $\$0.15$ .
So, if a letter weighs $x$ ounces, we have $x - 1$ additional ounces after the first ounce.
The cost of these additional ounces is $\$0.15 \cdot \lfloor x - 1 \rfloor$ if $x > 1$ .
If $x \leq 1$ , then $\lfloor x - 1 \rfloor$ will be zero or negative, but since we don't get money back for a letter weighing less than one ounce, we can still use this formula.

STEP 9

Putting it all together, the total cost $y$ is the initial cost of $\$0.55$ plus the additional cost of $\$0.15$ per ounce after the first ounce: $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$

STEP 10

Let's test with a weight of $x = 1$ ounce: $y = 0.55 + 0.15 \cdot \lfloor 1 - 1 \rfloor = 0.55 + 0.15 \cdot \lfloor 0 \rfloor = 0.55 + 0.15 \cdot 0 = 0.55$ That's $\$0.55$ , which is correct!

STEP 11

Now, let's test with $x = 2.5$ ounces: $y = 0.55 + 0.15 \cdot \lfloor 2.5 - 1 \rfloor = 0.55 + 0.15 \cdot \lfloor 1.5 \rfloor = 0.55 + 0.15 \cdot 1 = 0.55 + 0.15 = 0.70$ That's $\$0.70$ , which is $\$0.55$ for the first ounce and $\$0.15$ for the additional ounce, so it works!

STEP 12

The equation that represents the situation is $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$ , which is equivalent to the provided option $y=0.15\lfloor x-1\rfloor + 0.5$ .

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Studdy Solution

STEP 1

What is this asking? We need to find an equation that tells us the cost of mailing a letter based on its weight. Watch out! The first ounce has a different price than the rest, so be careful!

STEP 2

1. Define variables
2. Build the equation
3. Test the equation

STEP 3

Let's use $y$ to represent the total cost in $\$$ of mailing the letter.

STEP 4

Let's use $x$ to represent the weight of the letter in ounces.

STEP 5

The first ounce costs $\$0.55$ .
This is our starting point.

STEP 6

Each additional ounce after the first ounce costs $\$0.15$ .

STEP 7

We can use the greatest integer function (also known as the floor function), written as $\lfloor x \rfloor$ , which gives us the largest integer less than or equal to $x$ .
For example, $\lfloor 2.3 \rfloor = 2$ and $\lfloor 1 \rfloor = 1$ .

STEP 8

STEP 9

Putting it all together, the total cost $y$ is the initial cost of $\$0.55$ plus the additional cost of $\$0.15$ per ounce after the first ounce: $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$

STEP 10

Let's test with a weight of $x = 1$ ounce: $y = 0.55 + 0.15 \cdot \lfloor 1 - 1 \rfloor = 0.55 + 0.15 \cdot \lfloor 0 \rfloor = 0.55 + 0.15 \cdot 0 = 0.55$ That's $\$0.55$ , which is correct!

STEP 11

Now, let's test with $x = 2.5$ ounces: $y = 0.55 + 0.15 \cdot \lfloor 2.5 - 1 \rfloor = 0.55 + 0.15 \cdot \lfloor 1.5 \rfloor = 0.55 + 0.15 \cdot 1 = 0.55 + 0.15 = 0.70$ That's $\$0.70$ , which is $\$0.55$ for the first ounce and $\$0.15$ for the additional ounce, so it works!

STEP 12

The equation that represents the situation is $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$ , which is equivalent to the provided option $y=0.15\lfloor x-1\rfloor + 0.5$ .

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Studdy Solution

STEP 1

What is this asking? We need to find an equation that tells us the cost of mailing a letter based on its weight. Watch out! The first ounce has a different price than the rest, so be careful!

STEP 2

1. Define variables2. Build the equation3. Test the equation

STEP 3

Let's use yyy to represent the **total cost** in $\$$ of mailing the letter.

STEP 4

Let's use xxx to represent the **weight** of the letter in ounces.

STEP 5

The **first ounce** costs $0.55\$0.55$0.55.This is our **starting point**.

STEP 6

Each **additional ounce** *after* the first ounce costs $0.15\$0.15$0.15.

STEP 7

We can use the greatest integer function (also known as the floor function), written as ⌊x⌋\lfloor x \rfloor⌊x⌋, which gives us the largest integer *less than or equal to* xxx.For example, ⌊2.3⌋=2\lfloor 2.3 \rfloor = 2⌊2.3⌋=2 and ⌊1⌋=1\lfloor 1 \rfloor = 1⌊1⌋=1.

STEP 8

STEP 9

Putting it all together, the **total cost** yyy is the **initial cost** of $0.55\$0.55$0.55 plus the **additional cost** of $0.15\$0.15$0.15 per ounce after the first ounce: y=0.55+0.15⋅⌊x−1⌋ y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor y=0.55+0.15⋅⌊x−1⌋

STEP 10

STEP 11

STEP 12

The equation that represents the situation is y=0.55+0.15⋅⌊x−1⌋ y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor y=0.55+0.15⋅⌊x−1⌋, which is equivalent to the provided option y=0.15⌊x−1⌋+0.5y=0.15\lfloor x-1\rfloor + 0.5y=0.15⌊x−1⌋+0.5.

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Start learning now

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1. Define variables
2. Build the equation
3. Test the equation

Let's use $y$ to represent the total cost in $\$$ of mailing the letter.

Let's use $x$ to represent the weight of the letter in ounces.

The first ounce costs $\$0.55$ .
This is our starting point.

Each additional ounce after the first ounce costs $\$0.15$ .

We can use the greatest integer function (also known as the floor function), written as $\lfloor x \rfloor$ , which gives us the largest integer less than or equal to $x$ .
For example, $\lfloor 2.3 \rfloor = 2$ and $\lfloor 1 \rfloor = 1$ .

Putting it all together, the total cost $y$ is the initial cost of $\$0.55$ plus the additional cost of $\$0.15$ per ounce after the first ounce: $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$

The equation that represents the situation is $y = 0.55 + 0.15 \cdot \lfloor x - 1 \rfloor$ , which is equivalent to the provided option $y=0.15\lfloor x-1\rfloor + 0.5$ .