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 is already filled with 10 gallons, and it is continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have an equal amount of liquid. Use x

Question

x
as your variable.

Answer 1

Let's denote the amount of liquid in the first vat as \( A \) and the amount of liquid in the second vat as \( B \).

1. The first vat is initially empty and is filling at a rate of 6 gallons per minute. Therefore, the amount of liquid in the first vat after \( x \) minutes can be represented by: \[ A = 6x \]

2. The second vat starts with 10 gallons and is filling at a rate of 4 gallons per minute. Therefore, the amount of liquid in the second vat after \( x \) minutes can be represented by: \[ B = 10 + 4x \]

To find out when the two vats will have an equal amount of liquid, we set the two equations equal to each other: \[ 6x = 10 + 4x \]

Now, we can solve for \( x \).

1. Subtract \( 4x \) from both sides: \[ 6x - 4x = 10 \] \[ 2x = 10 \]

2. Divide both sides by 2: \[ x = 5 \]

Thus, the two vats will have an equal amount of liquid after 5 minutes. The equation representing when the two vats have equal amounts of liquid is: \[ 6x = 10 + 4x \]

Answer 2

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
x
as your variable.(1 point)

Answer 3

To compare the two participation plans, we can set up equations for each plan based on the number of visits per month, denoted by \( x \).

1. Cost of the first plan (paying $8 per visit): \[ \text{Cost}_1 = 8x \]

2. Cost of the second plan (paying $40 per month plus $2 per visit): \[ \text{Cost}_2 = 40 + 2x \]

To find the number of visits per month at which the costs of the two plans are equal, we set the two equations equal to each other: \[ 8x = 40 + 2x \]

Now, we can solve for \( x \):

1. Subtract \( 2x \) from both sides: \[ 8x - 2x = 40 \] \[ 6x = 40 \]

2. Divide both sides by 6: \[ x = \frac{40}{6} = \frac{20}{3} \approx 6.67 \]

Thus, the two plans will cost the same when there are approximately \( \frac{20}{3} \) visits (or about 6.67 visits) in a month. The equation representing the point where the costs are equal is: \[ 8x = 40 + 2x \]

Answer 4

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
x
as your variable.(1 point)

Answer 5

To find out when the two milkshake machines will have the same amount of milkshake left, we will set up equations for the remaining milkshake in each machine based on the number of days, denoted by \( x \).

1. Vanilla milkshake machine:

   • Initial capacity: 300 cups
   • Sells 15 cups per day
   • Amount left after \( x \) days: \[ V = 300 - 15x \]

2. Chocolate milkshake machine:

   • Initial capacity: 280 cups
   • Sells 20 cups per day
   • Amount left after \( x \) days: \[ C = 280 - 20x \]

To find when the two machines will have the same amount of milkshake left, set the two equations equal to each other: \[ 300 - 15x = 280 - 20x \]

Now, we can solve for \( x \):

1. Add \( 20x \) to both sides: \[ 300 - 15x + 20x = 280 \] \[ 300 + 5x = 280 \]

2. Subtract 300 from both sides: \[ 5x = 280 - 300 \] \[ 5x = -20 \]

3. Divide by 5: \[ x = -4 \]

Thus, the equation representing the number of days when the two machines will have the same amount of milkshake left is: \[ 300 - 15x = 280 - 20x \]

However, since \( x = -4 \), this means that 4 days before they started selling milkshakes, the two machines had the same amount of milkshake left, indicating that the vanilla machine has a larger initial capacity and sells less per day compared to the chocolate machine.

Answer 6

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.

Answer 7

Let's denote the amount of liquid in the first vat as \( A \) and the amount of liquid in the second vat as \( B \).

1. First vat: Initially empty and filling at a rate of 6 gallons per minute. The amount of liquid in the first vat after \( x \) minutes is given by: \[ A = 6x \]

2. Second vat: Already contains 28 gallons and filling at a rate of 4 gallons per minute. The amount of liquid in the second vat after \( x \) minutes is given by: \[ B = 28 + 4x \]

To find out when the two vats will have the same amount of liquid, we set the two equations equal to each other: \[ 6x = 28 + 4x \]

Now, we can solve for \( x \):

1. Subtract \( 4x \) from both sides: \[ 6x - 4x = 28 \] \[ 2x = 28 \]

2. Divide both sides by 2: \[ x = 14 \]

Thus, the two vats will have the same amount of liquid after 14 minutes.

Answer 8

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?

Answer 9

To compare the two participation plans, we can set up equations for the total cost based on the number of visits, denoted by \( x \).

1. Cost of the first plan (paying $10 per visit): \[ \text{Cost}_1 = 10x \]

2. Cost of the second plan (paying $30 per month plus $4 per visit): \[ \text{Cost}_2 = 30 + 4x \]

To find the number of visits at which the costs of the two plans are equal, we set the two equations equal to each other: \[ 10x = 30 + 4x \]

Now, we can solve for \( x \):

1. Subtract \( 4x \) from both sides: \[ 10x - 4x = 30 \] \[ 6x = 30 \]

2. Divide both sides by 6: \[ x = \frac{30}{6} = 5 \]

Therefore, the number of visits that would make the cost of both plans the same is 5 visits.

Answer 10

5 out of 5. Thank you

Answer 11

You're welcome! I'm glad I could help. If you have any more questions or need assistance with anything else, feel free to ask. Have a great day!