Math / Algebra

QuestionMiss Morton is getting married soon. She would like to hire a professional who will come to her house and do hair and makeup for the bridal party on her wedding day. She has narrowed it down to two possible choices. A woman from Riverside Salon charges an initial fee of $\$ 66$ and then an additional $\$ 22$ per hour. Alternatively, a team from Beauty by Frank offers a free consultation and then charges a higher hourly rate of $\$ 33$ . If it takes a certain amount of time to finish everyone's hair and makeup, Miss Morton will end up paying the same amount either way. How long would that take? How much would Miss Morton end up paying?
If it takes $\square$ hours, to finish everyone's hair and makeup, Miss Morton will end up paying \\square$

Studdy Solution

STEP 1

What is this asking? We need to find out how many hours it would take for two different hair and makeup services to cost the same and then figure out that total cost. Watch out! Don't get tricked by the free consultation, it just means there's no initial fee for Beauty by Frank!

STEP 2

1. Define the cost functions
2. Set the cost functions equal
3. Solve for the time
4. Calculate the total cost

STEP 3

Let's define the cost function for Riverside Salon.
We'll call it $R(t)$ , where $t$ is the time in hours.
The initial fee is $\$66$ , and the hourly rate is $\$22$ .
So, the cost is the initial fee plus the hourly rate times the number of hours: $R(t) = 66 + 22 \cdot t$

STEP 4

Now, let's define the cost function for Beauty by Frank.
We'll call it $B(t)$ , where $t$ is the time in hours.
There's no initial fee, and the hourly rate is $\$33$ .
So, the cost is simply the hourly rate times the number of hours: $B(t) = 33 \cdot t$

STEP 5

We want to find the time when the costs are equal, so we set the two cost functions equal to each other: $R(t) = B(t)$ $66 + 22 \cdot t = 33 \cdot t$

STEP 6

To isolate $t$ , we'll subtract $22 \cdot t$ from both sides of the equation: $66 + 22 \cdot t - 22 \cdot t = 33 \cdot t - 22 \cdot t$ $66 = 11 \cdot t$

STEP 7

Now, we'll divide both sides by $11$ to find the time: $\frac{66}{11} = \frac{11 \cdot t}{11}$ $6 = t$ So, it would take 6 hours for the costs to be the same.

STEP 8

Now that we know the time, we can plug it back into either cost function to find the total cost.
Let's use $R(t)$ : $R(6) = 66 + 22 \cdot 6$ $R(6) = 66 + 132$ $R(6) = 198$ So, Miss Morton would end up paying $\$198$ .

STEP 9

If it takes 6 hours to finish everyone's hair and makeup, Miss Morton will end up paying $\$ \textbf{198}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find out how many hours it would take for two different hair and makeup services to cost the same and then figure out that total cost. Watch out! Don't get tricked by the free consultation, it just means there's no initial fee for Beauty by Frank!

STEP 2

1. Define the cost functions
2. Set the cost functions equal
3. Solve for the time
4. Calculate the total cost

STEP 3

Let's define the cost function for Riverside Salon.
We'll call it $R(t)$ , where $t$ is the time in hours.
The initial fee is $\$66$ , and the hourly rate is $\$22$ .
So, the cost is the initial fee plus the hourly rate times the number of hours: $R(t) = 66 + 22 \cdot t$

STEP 4

Now, let's define the cost function for Beauty by Frank.
We'll call it $B(t)$ , where $t$ is the time in hours.
There's no initial fee, and the hourly rate is $\$33$ .
So, the cost is simply the hourly rate times the number of hours: $B(t) = 33 \cdot t$

STEP 5

We want to find the time when the costs are equal, so we set the two cost functions equal to each other: $R(t) = B(t)$ $66 + 22 \cdot t = 33 \cdot t$

STEP 6

To isolate $t$ , we'll subtract $22 \cdot t$ from both sides of the equation: $66 + 22 \cdot t - 22 \cdot t = 33 \cdot t - 22 \cdot t$ $66 = 11 \cdot t$

STEP 7

Now, we'll divide both sides by $11$ to find the time: $\frac{66}{11} = \frac{11 \cdot t}{11}$ $6 = t$ So, it would take 6 hours for the costs to be the same.

STEP 8

Now that we know the time, we can plug it back into either cost function to find the total cost.
Let's use $R(t)$ : $R(6) = 66 + 22 \cdot 6$ $R(6) = 66 + 132$ $R(6) = 198$ So, Miss Morton would end up paying $\$198$ .

STEP 9

If it takes 6 hours to finish everyone's hair and makeup, Miss Morton will end up paying $\$ \textbf{198}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find out how many hours it would take for two different hair and makeup services to cost the same and then figure out that total cost. Watch out! Don't get tricked by the free consultation, it just means there's no initial fee for Beauty by Frank!

STEP 2

1. Define the cost functions2. Set the cost functions equal3. Solve for the time4. Calculate the total cost

STEP 3

STEP 4

Now, let's **define** the cost function for Beauty by Frank.We'll call it B(t)B(t)B(t), where ttt is the time in hours.There's no initial fee, and the **hourly rate** is $33\$33$33.So, the cost is simply the hourly rate times the number of hours: B(t)=33⋅tB(t) = 33 \cdot tB(t)=33⋅t

STEP 5

We want to find the time when the costs are equal, so we set the two cost functions equal to each other: R(t)=B(t)R(t) = B(t)R(t)=B(t) 66+22⋅t=33⋅t66 + 22 \cdot t = 33 \cdot t66+22⋅t=33⋅t

STEP 6

To **isolate** ttt, we'll **subtract** 22⋅t22 \cdot t22⋅t from both sides of the equation: 66+22⋅t−22⋅t=33⋅t−22⋅t66 + 22 \cdot t - 22 \cdot t = 33 \cdot t - 22 \cdot t66+22⋅t−22⋅t=33⋅t−22⋅t 66=11⋅t66 = 11 \cdot t66=11⋅t

STEP 7

Now, we'll **divide** both sides by 111111 to find the **time**: 6611=11⋅t11\frac{66}{11} = \frac{11 \cdot t}{11}1166​=1111⋅t​ 6=t6 = t6=tSo, it would take **6 hours** for the costs to be the same.

STEP 8

Now that we know the time, we can plug it back into either cost function to find the total cost.Let's use R(t)R(t)R(t): R(6)=66+22⋅6R(6) = 66 + 22 \cdot 6R(6)=66+22⋅6 R(6)=66+132R(6) = 66 + 132R(6)=66+132R(6)=198R(6) = 198R(6)=198So, Miss Morton would end up paying $198\$198$198.

STEP 9

If it takes **6** hours to finish everyone's hair and makeup, Miss Morton will end up paying $198\$ \textbf{198}$198.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the cost functions
2. Set the cost functions equal
3. Solve for the time
4. Calculate the total cost

Now, let's define the cost function for Beauty by Frank.
We'll call it $B(t)$ , where $t$ is the time in hours.
There's no initial fee, and the hourly rate is $\$33$ .
So, the cost is simply the hourly rate times the number of hours: $B(t) = 33 \cdot t$

We want to find the time when the costs are equal, so we set the two cost functions equal to each other: $R(t) = B(t)$ $66 + 22 \cdot t = 33 \cdot t$

To isolate $t$ , we'll subtract $22 \cdot t$ from both sides of the equation: $66 + 22 \cdot t - 22 \cdot t = 33 \cdot t - 22 \cdot t$ $66 = 11 \cdot t$

Now, we'll divide both sides by $11$ to find the time: $\frac{66}{11} = \frac{11 \cdot t}{11}$ $6 = t$ So, it would take 6 hours for the costs to be the same.

Now that we know the time, we can plug it back into either cost function to find the total cost.
Let's use $R(t)$ : $R(6) = 66 + 22 \cdot 6$ $R(6) = 66 + 132$ $R(6) = 198$ So, Miss Morton would end up paying $\$198$ .

If it takes 6 hours to finish everyone's hair and makeup, Miss Morton will end up paying $\$ \textbf{198}$ .