Courtney organized a charity walk. In the first year, the walk generated $3000. She hopes to increase this amount by $900 each year for the next several years. If her goal is met, in how many years will the walk have generated a total of at least $65,000 for the charity?
Well, Courtney seems to be really ambitious with this charity walk! Let's do some math here.
Since Courtney hopes to increase the amount by $900 each year, we can create an equation to represent the total amount generated after "n" years:
Total amount generated = $3000 + $900n
Now, we need to find the number of years it takes for the total amount to reach at least $65,000. Let's set up the equation:
$65,000 ≤ $3000 + $900n
Now let's solve for "n":
$900n ≥ $65,000 - $3000
$900n ≥ $62,000
Dividing both sides of the equation by $900:
n ≥ $62,000 / $900
n ≥ 68.88
As we can't have a fraction of a year, we round up to the nearest whole number. Therefore, it will take Courtney at least 69 years for the walk to generate at least $65,000 for the charity.
Good luck to Courtney, and may her future challenges be "well-rounded" – just like her fundraising efforts!
To find out how many years it will take for the walk to generate at least $65,000, we can set up an equation.
Let's call the number of years it takes to reach the goal "n."
The amount generated in the first year is $3000.
Since the goal is to increase the amount by $900 each year, the amount generated in the second year will be $3000 + $900 = $3900.
In the third year, the amount generated will be $3900 + $900 = $4800.
We can see that the amount generated each year follows an arithmetic sequence with a common difference of $900.
The formula to find the sum of an arithmetic sequence is:
Sn = (n/2) * (2a + (n-1)d)
Where Sn is the sum of the n terms, a is the first term, and d is the common difference.
We can use this formula to set up an equation:
65,000 = (n/2) * (2 * 3000 + (n-1) * 900)
Simplifying the equation:
65,000 = n * (6000 + 900n - 900)
65,000 = n * (6000 - 900 + 900n)
65,000 = n * (5100 + 900n)
65,000 = 5100n + 900n^2
Rearranging and setting the equation to 0:
900n^2 + 5100n - 65,000 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not so simple, so we will use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula:
n = (-5100 ± √(5100^2 - 4 * 900 * -65,000)) / (2 * 900)
Calculating:
n = (-5100 ± √(26010000 + 23400000)) / 1800
n = (-5100 ± √(49410000)) / 1800
n = (-5100 ± 7020.055) / 1800
We have two possible solutions:
n = (-5100 + 7020.055) / 1800 ≈ 7.51
or
n = (-5100 - 7020.055) / 1800 ≈ -2.08
Since it doesn't make sense for the number of years to be negative, we discard the second solution.
Therefore, Courtney's charity walk will have generated at least $65,000 in approximately 7.51 years.
To solve this problem, we need to find out how many years it will take for the walk to generate a total of at least $65,000 for the charity.
Let's break down the problem step by step:
1. In the first year, the walk generated $3000.
2. Courtney hopes to increase this amount by $900 each year for the next several years.
3. We need to determine the number of years it will take for the total amount generated to reach at least $65,000.
To find the answer, we can set up an equation based on the information given:
Total amount generated = $3000 + ($900 per year) * (number of years)
Now, we need to solve for the number of years. Let's substitute the values into the equation:
$65,000 = $3000 + ($900 per year) * (number of years)
To isolate the variable (number of years), we can subtract $3000 from both sides:
$65,000 - $3000 = ($900 per year) * (number of years)
Simplifying the equation:
$62,000 = $900 * (number of years)
Next, divide both sides of the equation by $900:
$62,000 / $900 = number of years
Using a calculator:
Number of years ≈ 68.89
Since we can't have a fraction of a year, we need to round the number up to the nearest whole number. Therefore, it will take approximately 69 years for the walk to generate a total of at least $65,000 for the charity.
Increase per year = $900
Target=$65000
current amount = 3000
Number of years to generate exactly 65000
=(65000-3000)/900
= 68.89 years
So minimum number of years to equal or exceed target = 69 years
(a little more than "several" years!)