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 region that satisfies the given requirements, we can write the following system of inequalities:
1. The total amount donated must be less than or equal to $480:
x + y ≤ 480
2. The amount donated to the Educational Growth Foundation must be at least $140:
y ≥ 140
3. The amount donated to the Educational Growth Foundation must be at least twice the amount donated to the City Youth Fund:
y ≥ 2x
Now, let's graph these inequalities on a coordinate plane.
First, let's graph the first inequality, x + y ≤ 480:
To graph this, we'll draw a dashed line with a slope of -1 passing through (0, 480) and (480, 0), and shade below the line since we want x + y to be less than or equal to 480.
Next, let's graph the second inequality, y ≥ 140:
To graph this, we'll draw a solid horizontal line passing through y = 140 and shade above the line since we want y to be greater than or equal to 140.
Finally, let's graph the third inequality, y ≥ 2x:
To graph this, we'll draw a dashed line with a slope of 2 passing through (0, 0) and (240, 480), and shade above the line since we want y to be greater than or equal to 2x.
Combining all the shaded regions, we get a triangular region.
Please note that I cannot draw the region using this text-based format, but you can draw it on a graph paper or use graphing software by following the instructions above.
To determine the region corresponding to all values of "x" and "y" that satisfy the given requirements, we can set up a system of inequalities based on the given conditions.
Let's start by defining the variables:
x = the amount of money (in dollars) donated to the City Youth Fund
y = the amount of money (in dollars) donated to the Educational Growth Foundation
Now, let's establish the conditions:
1. Tammy will donate up to $480 to charity. This implies the total donation must be less than or equal to $480:
x + y ≤ 480
2. Tammy would like to donate at least $140 dollars to the Educational Growth Foundation. This implies the donation to the Educational Growth Foundation must be greater than or equal to $140:
y ≥ 140
3. Tammy would like the amount donated to the Educational Growth Foundation to be at least twice the amount donated to the City Youth Fund. This implies that the donation to the Educational Growth Foundation must be greater than or equal to twice the amount of the City Youth Fund:
y ≥ 2x
By representing these conditions as inequalities, we have:
x + y ≤ 480
y ≥ 140
y ≥ 2x
We can graph these inequalities on a coordinate plane to visualize the region that satisfies all these conditions. However, since I can't generate a graph here, I will describe the region to you:
Firstly, draw a line with a slope of 2 passing through the point (0,0) on the coordinate plane. This represents the equation y = 2x.
Next, shade the region above this line, as it satisfies the condition y ≥ 2x.
Then, draw a horizontal line at y = 140. Shade the region above this line as well, as it satisfies the condition y ≥ 140.
Finally, draw the line x + y = 480. Shade the region below this line since it satisfies the condition x + y ≤ 480.
The shaded region where all three conditions are satisfied represents the possible values of x and y that meet the requirements.