The amount of paint an electric paint sprayer applies varies directly with time. A sprayer is set to apply 0.5 gallon in 2.5 minutes. Write an equation that gives the amount P of paint as a function of the time T. How much paint is applied if the sprayer is operated for 20 minutes.
Well, since the amount of paint applied varies directly with time, we can use the equation P = kT, where P is the amount of paint applied, T is the time in minutes, and k is the constant of variation.
In this case, we know that when T = 2.5 minutes, P = 0.5 gallon. So we can substitute these values into the equation to find k:
0.5 = k * 2.5
Solving for k, we get k = 0.2.
Now we can use this value of k to find the amount of paint applied if the sprayer is operated for 20 minutes:
P = 0.2 * 20
P = 4 gallons.
So if you operate the sprayer for 20 minutes, approximately 4 gallons of paint will be applied. Just be careful not to go overboard and turn your walls into abstract art installations!
To write an equation that gives the amount P of paint as a function of the time T, we can use the concept of direct variation. Direct variation means that as one variable (in this case, time T) increases, the other variable (amount P of paint) increases in proportion.
Let's first identify the constant of variation, which represents the rate at which the paint is being applied. In this case, we know that the paint sprayer applies 0.5 gallon in 2.5 minutes. So, the constant of variation is 0.5 gallon/2.5 minutes.
Now, we can set up the equation:
P = k * T
where P represents the amount of paint, T represents the time in minutes, and k is the constant of variation.
Since we know that when T is 2.5 minutes, P is 0.5 gallon, we can substitute these values into the equation:
0.5 = k * 2.5
To solve for k, divide 0.5 by 2.5:
k = 0.5/2.5
k = 0.2
Now we can rewrite the equation as:
P = 0.2 * T
To find out how much paint is applied if the sprayer is operated for 20 minutes, we can substitute T = 20 into the equation:
P = 0.2 * 20
P = 4 gallons
Therefore, if the sprayer is operated for 20 minutes, 4 gallons of paint will be applied.