A yoga studio has two participation plans. For the first plan, the cost is $8 per visit. For the second plan, the cost is $40 per month, and $2 per visit. Write an equation to show the number of visits per month that it would take for the two plans to be equal. Use x

Question

as your variable.

A diner makes vanilla milkshakes and chocolate milkshakes. The vanilla milkshake machine has a 300-cup capacity, and sells 15 cups per day. The chocolate milkshake machine has a 280-cup capacity, and sells 20 cups per day. Write an equation to show when the two machines will have the same amount of milkshake left. Use x
as your variable.(1 point)

Two pumps are filling large vats of liquid. One vat is empty, and the pump is filling it at a rate of 6 gallons per minute. The other vat already contains 28 gallons and is continuing to fill at a rate of 4 gallons per minute. When will the two vats have the same amount of liquid?(1 point)
They will have the same amount of liquid after
minutes.

A yoga studio has two participation plans. The first plan costs $10 per visit. The second plan costs $30 per month, and $4 per visit. How many visits would make the cost of both plans the same?(1 point)
The plans would cost the same at
visits.

Answer 1

Let's tackle each problem one by one, starting with the yoga studio participation plans.

1. Yoga Studio Participation Plans:

Plan 1: $8 per visit
Plan 2: $40 per month + $2 per visit

Let \( x \) be the number of visits in a month. We need to set up an equation to determine when the costs of the two plans are equal.

Cost of Plan 1:
\[ C_1 = 8x \]

Cost of Plan 2:
\[ C_2 = 40 + 2x \]

Now, set the two costs equal to each other: \[ 8x = 40 + 2x \]

2. Diner's Milkshake Machines:

Vanilla Milkshake Machine:
Capacity = 300 cups, selling 15 cups per day.

Chocolate Milkshake Machine:
Capacity = 280 cups, selling 20 cups per day.

Let \( x \) be the number of days.

Remaining Vanilla Milkshakes after \( x \) days:
\[ R_v = 300 - 15x \]

Remaining Chocolate Milkshakes after \( x \) days:
\[ R_c = 280 - 20x \]

We set the amounts equal: \[ 300 - 15x = 280 - 20x \]

3. Two Pumps Filling Vats:

Pump 1: Filling at 6 gallons per minute from empty vat.
Pump 2: Already has 28 gallons and filling at 4 gallons per minute.

Let \( x \) be the number of minutes.

Liquid in Vat 1 after \( x \) minutes:
\[ L_1 = 6x \]

Liquid in Vat 2 after \( x \) minutes:
\[ L_2 = 28 + 4x \]

Set them equal: \[ 6x = 28 + 4x \]

4. Yoga Studio Participation Plans (Second Case):

Plan 1: $10 per visit
Plan 2: $30 per month + $4 per visit

Let \( x \) be the number of visits.

Cost of Plan 1:
\[ C_1 = 10x \]

Cost of Plan 2:
\[ C_2 = 30 + 4x \]

Setting them equal: \[ 10x = 30 + 4x \]

Summary of Answers:

1. For the first yoga studio plan, the equation is: \[ 8x = 40 + 2x \]

2. For the diner milkshake machines, the equation is: \[ 300 - 15x = 280 - 20x \]

3. For the two pumps filling vats, they will have the same amount after solving: \[ 6x = 28 + 4x \]

4. For the yoga studio second case, the equation is: \[ 10x = 30 + 4x \]

Now, let's calculate the number of visits or the time when the conditions are met:

Solving Each Equation:

1. First Plan:
   \[ 8x - 2x = 40 \]
   \[ 6x = 40 \]
   \[ x = \frac{40}{6} = \frac{20}{3} \approx 6.67 \text{ visits} \]

2. Diner's Milkshake Machines:
   \[ 300 - 15x = 280 - 20x \]
   \[ 20x - 15x = 300 - 280 \]
   \[ 5x = 20 \]
   \[ x = 4 \text{ days} \]

3. Two Pumps:
   \[ 6x - 4x = 28 \]
   \[ 2x = 28 \]
   \[ x = 14 \text{ minutes} \]

4. Second Plan:
   \[ 10x - 4x = 30 \]
   \[ 6x = 30 \]
   \[ x = 5 \text{ visits} \]

Final Answers

• Visits to equal cost for the first plan: Approximately 6.67 visits.
• Days for milkshake machines to have the same amount: 4 days.
• Minutes for the two vats to have the same amount: 14 minutes.
• Visits to equal cost for the second plan: 5 visits.