A small pool is being drained. There are 1500 gallons of water remaining in the pool after 5 minutes and 1000 gallons after 10 minutes. How much water was in the pool BEFORE it started draining.
Let's call the initial amount of water in the pool "x".
We know that after 5 minutes, there are 1500 gallons of water remaining. This means that in the first 5 minutes, (x - 1500) gallons of water were drained.
Similarly, after 10 minutes, there are 1000 gallons of water remaining. This means that in the next 5 minutes (from 5 to 10 minutes), another (1500 - 1000) = 500 gallons were drained.
So in total, (x - 1500) + 500 = (x - 1000) gallons of water were drained in 10 minutes.
Simplifying this equation, we get:
x - 1000 = (x - 1500) + 500
x - 1000 = x - 1000
Solving for x, we get:
x = 2000
So there were originally 2000 gallons of water in the pool before it started draining.
The first five terms of an arithmetic sequence are 29, 25.5, 22, 18.5, and 15. Which function, f(n), could be used to describe the nth term of the sequence?
Let's first find the common difference, d, between consecutive terms in this arithmetic sequence:
d = 25.5 - 29 = -3.5
d = 22 - 25.5 = -3.5
d = 18.5 - 22 = -3.5
d = 15 - 18.5 = -3.5
We see that the common difference is -3.5 for this arithmetic sequence. Now we can use this to find the general formula for the nth term using the formula:
a_n = a_1 + (n - 1)d
where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
Using the given first term, a_1 = 29, and the common difference, d = -3.5, we get:
a_n = 29 + (n - 1)(-3.5)
Simplifying and rearranging, we get:
a_n = 32.5 - 3.5n
Therefore, the function f(n) that could be used to describe the nth term of the sequence is:
f(n) = 32.5 - 3.5n
To determine the amount of water in the pool before it started draining, we can set up a linear equation using the given information.
Let's assume that the initial amount of water in the pool is represented by "x" gallons.
According to the given information, after 5 minutes there are 1500 gallons of water remaining, and after 10 minutes, there are 1000 gallons of water remaining.
We can set up the following equation: x - 5k = 1500, where "k" represents the draining rate in gallons per minute.
We can also set up the equation: x - 10k = 1000.
Now, we have a system of two linear equations:
x - 5k = 1500
x - 10k = 1000
We can solve this system of equations using substitution, elimination, or other methods. Let's use the elimination method to solve this system:
Multiply the second equation by 2 to eliminate "x":
2(x - 10k) = 2(1000)
2x - 20k = 2000
Now, subtract the first equation from the new equation:
(2x - 20k) - (x - 5k) = 2000 - 1500
x - 15k = 500
Now we have a single equation that we can solve for "x". Rearrange the equation:
x = 500 + 15k
Since we cannot determine the precise value of "k" from the given information, we cannot calculate the exact value of "x". However, we do know that the initial amount of water in the pool, represented by "x", is 500 gallons more than 15 times the draining rate, "k".