According to a survey conducted in 1990 by
Independent Sector, the percent of their incomes
that Americans give to charities is related
to their household incomes. For families
with annual incomes between $5000 and
$100,000, the percent is modeled by
P = 0.0014x^2 − 0.1529 x + 5.855
Where P is the percentage of annual income
given and x is the annual income in thousands of dollars.
What is the largest of the two annual incomes
at which Americans give 3.2%(P = 3.2) of their income to charity?
When P = 3.2,
3.2 = 0.0014x^2 − 0.1529x + 5.855
=> 0.0014x^2 − 0.1529 x + 2.655 = 0
Solving the quadratic, we get:
x = 87.554, x = 21.660
Since x in is thousands, the two amounts are:
$87,554 and $21,660
Which does the question ask for?
To find the largest of the two annual incomes at which Americans give 3.2% of their income to charity, we need to solve the equation P = 3.2 for x.
1. Substitute 3.2 for P in the equation:
3.2 = 0.0014x^2 - 0.1529x + 5.855
2. Rearrange the equation to a quadratic form:
0.0014x^2 - 0.1529x + 5.855 - 3.2 = 0
3. Simplify the equation:
0.0014x^2 - 0.1529x + 2.655 = 0
4. Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 0.0014, b = -0.1529, and c = 2.655.
5. Substitute the values into the quadratic formula:
x = (0.1529 ± √((-0.1529)^2 - 4 * 0.0014 * 2.655)) / (2 * 0.0014)
6. Simplify under the square root:
x = (0.1529 ± √(0.02341441 - 0.0185772)) / 0.0028
x = (0.1529 ± √0.00483721) / 0.0028
x = (0.1529 ± 0.0695839) / 0.0028
7. Solve for x:
x1 = (0.1529 + 0.0695839) / 0.0028
= 0.2224839 / 0.0028
≈ 79.45
x2 = (0.1529 - 0.0695839) / 0.0028
= 0.0833161 / 0.0028
≈ 29.75
The two solutions for x are approximately 79.45 and 29.75. Since we are looking for the largest annual income, the largest value is approximately $79,450 (79.45 * 1000).
To find the largest of the two annual incomes at which Americans give 3.2% of their income to charity, we need to solve the equation P = 3.2 and find the corresponding value of x.
Given that the equation relating the percentage of annual income given to charity (P) to the annual income in thousands of dollars (x) is P = 0.0014x^2 − 0.1529x + 5.855, we can set this equation equal to 3.2:
0.0014x^2 − 0.1529x + 5.855 = 3.2
Now, let's solve this quadratic equation to find the values of x.
First, we need to rearrange the equation and set it equal to zero:
0.0014x^2 − 0.1529x - 1.655 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 0.0014, b = -0.1529, and c = -1.655. Plugging these values into the formula, we get:
x = (-(-0.1529) ± √((-0.1529)^2 - 4(0.0014)(-1.655))) / 2(0.0014)
Simplifying further:
x = (0.1529 ± √(0.02339541 + 0.0092432)) / 0.0028
x = (0.1529 ± √0.03263861) / 0.0028
Taking the square root:
x = (0.1529 ± 0.1804765591) / 0.0028
Now, we'll calculate both solutions:
x1 = (0.1529 + 0.1804765591) / 0.0028 ≈ 109.805
x2 = (0.1529 - 0.1804765591) / 0.0028 ≈ -18.274
Since we are interested in the largest of the two annual incomes, we discard the negative value. Therefore, the largest annual income at which Americans give 3.2% of their income to charity is approximately $109,805.