Ari begins painting at 12:00 noon. At 12:30 he estimates that 15.75 gallons of paint are left, and at 2:00 he estimates that 10.5 gallons of paint remain. If the paint is used at a constant rate, how many gallons of paint were there at the start of the job?
I feel dumb for not knowing how to do this, but I really need help.
Thank you!
he used 15.75-10.5 = 5.25 gallons of paint in 1.5 hours. That is 3.5 gal/hr
Thus, using the point-slope form of the line, if y is the amount of paint left at x hours after 12:00, then
y-10.5 = -3.5(x-2)
So, find y when x=0.
Or, knowing that he used 3.5 gal/hr, and knowing that he had 15.75 gal 1/2 hr after starting, just add half of 3.5 to 15.75.
Either way, you will find he started with 17.5 gal of paint.
Ok, or I could find the slope correct? Thank you!
No need to feel dumb! I'm here to help you out. Let's solve this step-by-step:
Step 1: Determine the time interval between 12:30 and 2:00.
The time interval is 2:00 - 12:30, which is 1.5 hours.
Step 2: Calculate the rate of paint usage.
The rate of paint usage is the difference in gallons of paint divided by the time interval.
The difference in gallons of paint is 15.75 gallons - 10.5 gallons = 5.25 gallons.
The rate of paint usage is 5.25 gallons / 1.5 hours = 3.5 gallons per hour.
Step 3: Calculate the amount of paint used from 12:00 to 12:30.
Since 30 minutes have passed (half an hour), the amount of paint used is half the rate of paint usage.
The amount of paint used from 12:00 to 12:30 is 0.5 * 3.5 gallons = 1.75 gallons.
Step 4: Subtract the paint used from the estimate at 12:30 to find the initial amount of paint.
The estimate at 12:30 was 15.75 gallons, and the amount used from 12:00 to 12:30 was 1.75 gallons.
Therefore, the initial amount of paint at 12:00 was 15.75 gallons + 1.75 gallons = 17.5 gallons.
So, there were initially 17.5 gallons of paint at the start of the job.
No need to feel dumb! I'm here to help you out. To solve this problem, we can use the information given and create a linear equation to find the initial amount of paint.
Let's break down the information we have:
1. Ari begins painting at 12:00 noon.
2. At 12:30, he estimates that 15.75 gallons of paint are left.
3. At 2:00, he estimates that 10.5 gallons of paint remain.
The time span between 12:00 and 12:30 is 0.5 hours, and the time span between 12:00 and 2:00 is 2 hours.
We can assume that the amount of paint used is constant during this time. So, the rate at which paint is used per hour is the same.
Now, let's set up an equation to represent this situation:
Let "x" represent the initial amount of paint.
Since the rate of paint used is constant, we can say that the amount of paint remaining at each estimated time is equal to the initial amount of paint minus the rate of paint used multiplied by the time elapsed.
At 12:30 (0.5 hours elapsed): Remaining paint = x - (rate * 0.5) = 15.75 gallons
At 2:00 (2 hours elapsed): Remaining paint = x - (rate * 2) = 10.5 gallons
To find the rate, we can subtract the two equations:
(x - (rate * 2)) - (x - (rate * 0.5)) = 10.5 - 15.75
Simplifying the equation:
x - 2 * rate - x + 0.5 * rate = -5.25
-1.5 * rate = -5.25
rate = -5.25 / (-1.5)
rate = 3.5 gallons per hour
Now that we have the rate, we can substitute it back into one of the equations to solve for x (the initial amount of paint):
x - (3.5 * 0.5) = 15.75
x - 1.75 = 15.75
x = 15.75 + 1.75
x = 17.5 gallons
Therefore, the initial amount of paint was 17.5 gallons.
I hope that helps! Let me know if you have any further questions.