Question

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 10 gallons, and it is continuing to fill at a rate of 4 gallons per minute. Writes an equation to show when the two vats will have an equal amount of liquis. Use x as your variable.

Answer 1

To compare the amounts of liquid in the two vats, we can define the variable \( x \) as the number of minutes that have passed since the pumps started filling the vats.

1. For the first vat (starting from empty and filling at 6 gallons per minute):

   • The amount of liquid in this vat after \( x \) minutes can be expressed as: \[ A_1 = 6x \]

2. For the second vat (starting with 10 gallons and filling at 4 gallons per minute):

   • The amount of liquid in this vat after \( x \) minutes can be expressed as: \[ A_2 = 10 + 4x \]

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

Now, we can solve for \( x \) to find the time when both vats contain the same amount of liquid.

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

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

So, the two vats will contain equal amounts of liquid after 5 minutes.

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. Writes an equation to show the number of visits per month that it would take the two plans to be equal. Use x as you variable.

Answer 3

To compare the costs of the two participation plans at a yoga studio, we will set up equations based on the number of visits, denoted by \( x \).

1. For the first plan (cost is $8 per visit): \[ C_1 = 8x \]

2. For the second plan (cost is $40 per month, plus $2 per visit): \[ C_2 = 40 + 2x \]

To find out when the two plans cost the same, we set the two cost expressions equal to each other: \[ 8x = 40 + 2x \]

Now, we solve for \( x \):

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

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

So, it would take approximately \( 6.67 \) visits per month for the two plans to have equal costs. Since you can't attend a fraction of a visit in reality, you would need to make at least 7 visits for the second plan to be more cost-effective.

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 milkshakes left. Use x as your variable.

Answer 5

To compare the amounts of milkshakes left in the two machines, we will set up equations based on the number of days passed, denoted by \( x \).

1. For the vanilla milkshake machine:

   • The starting capacity is 300 cups, and it sells 15 cups per day.
   • The amount of milkshakes left after \( x \) days can be expressed as: \[ V(x) = 300 - 15x \]

2. For the chocolate milkshake machine:

   • The starting capacity is 280 cups, and it sells 20 cups per day.
   • The amount of milkshakes left after \( x \) days can be expressed as: \[ C(x) = 280 - 20x \]

To find out when the amounts of milkshakes left in both machines will be equal, we set the two expressions equal to each other: \[ 300 - 15x = 280 - 20x \]

Now, let's solve for \( x \):

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

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

3. Finally, divide by 5: \[ x = \frac{-20}{5} = -4 \]

The result \( x = -4 \) indicates that the two machines had the same amount of milkshakes left 4 days before you started counting. This suggests that at that point in time, both machines had the same amount of milkshakes remaining, but they will diverge after that time given their sales rates.

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 it is continuing to fill at a rate of 4 gallons per minute. When will the two vats have the same amount of liquid?

Answer 7

To determine when the two vats will contain the same amount of liquid, we can set up equations based on time, denoted by \( x \), in minutes.

1. For the first vat (starting from empty and filling at 6 gallons per minute):

   • The amount of liquid in this vat after \( x \) minutes is: \[ A_1 = 6x \]

2. For the second vat (starting with 28 gallons and filling at 4 gallons per minute):

   • The amount of liquid in this vat after \( x \) minutes is: \[ A_2 = 28 + 4x \]

To find out when both vats have equal amounts of liquid, we set the two expressions equal to each other: \[ 6x = 28 + 4x \]

Now, let's solve for \( x \):

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

2. Divide both sides by 2: \[ x = \frac{28}{2} = 14 \]

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

Answer 8

A yoga studio has two participation plans. For the first plan, the cost is $10 per visit. For the second plan, the cost is $30 per month, and $4 per visit. How many visits would make the cost of both plans the same?

Answer 9

To find out how many visits would make the cost of both participation plans the same, we can set up equations based on the number of visits, denoted by \( x \).

1. For the first plan (cost is $10 per visit): \[ C_1 = 10x \]

2. For the second plan (cost is $30 per month, plus $4 per visit): \[ C_2 = 30 + 4x \]

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

Now, let's 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, 5 visits would make the cost of both plans the same.