Tammy will donate up to
$480
to charity. The money will be divided between two charities: the City Youth Fund and the Educational Growth Foundation. Tammy would like to donate at least
$140
dollars to the Educational Growth Foundation. She would also like the amount donated to the Educational Growth Foundation to be at least twice the amount donated to the City Youth Fund. Let
x
denote the amount of money (in dollars) donated to the City Youth Fund. Let
y
denote the amount of money (in dollars) donated to the Educational Growth Foundation. Shade the region corresponding to all values of
x
and
y
that satisfy these requirements.
To determine the values of x and y that satisfy the given requirements and shade the region, we can set up a system of inequalities based on the given information.
Let's break down the information provided:
1. Tammy will donate up to $480 to charity:
This means that the total amount donated, x + y, should be less than or equal to $480.
Mathematically, this can be expressed as: x + y ≤ 480.
2. Tammy wants to donate at least $140 to the Educational Growth Foundation:
This means that y should be greater than or equal to $140.
Mathematically, this can be expressed as: y ≥ 140.
3. The amount donated to the Educational Growth Foundation should be at least twice the amount donated to the City Youth Fund:
This means that y should be greater than or equal to 2x.
Mathematically, this can be expressed as: y ≥ 2x.
Now let's combine these inequalities to represent the shaded region:
x + y ≤ 480 ----------(1)
y ≥ 140 ------------(2)
y ≥ 2x --------------(3)
To graph this region, we can start by graphing the lines represented by the equalities:
For equation (1): x + y = 480, plot the line with a slope of -1 passing through the point (0, 480).
For equation (2): y = 140, plot the horizontal line at y = 140.
For equation (3): y = 2x, plot the line with a slope of 2 passing through the origin (0, 0).
Next, we need to determine which side of each line represents the shaded region. Since we want values of x and y that satisfy all three inequalities simultaneously, we need to find the region where all three shaded regions overlap.
Shade the region that satisfies all three inequalities, and this will be the desired region where x and y satisfy the given requirements.