You are planning an ornamental garden that has a total area of 200 square feet. There are two sizes of ornamental plants you have chosen for your garden. Each small plant requires 2.5 square feet of space and each large plant requires 8 square feet of space. This situation can be modeled by the equation where x is the number of small plants and y is the number of large plants to be placed in the garden.
1.What are 3 possibliities of small plants and large plants you can plant in the garden?
2.5x + 8y = 200
5x + 16y = 400
We can graph this on the x-y plane and only look what the line does in the first quadrant.
x-intercept, let y = 0 , x = 80
y intercept , let x = 0 , y = 25
slope of line = -25/80 = -5/16
So starting with the point (80,0)
we can decrease the x by 16 and increase the y by 5 to get a new point
---> (64, 5)
---> (48, 10)
---> (32, 15)
---> (16, 20)
---> (0, 25)
pick any 3
testing one of them:
e.g. (32,15)
32 small + 15 large
takes 32(2.5) + 15(8) = 200
To find three possible combinations of small and large plants that can be planted in the garden, we need to satisfy the given conditions of the total area.
Let x be the number of small plants and y be the number of large plants.
The area taken by small plants = 2.5x square feet.
The area taken by large plants = 8y square feet.
According to the problem, the total area of the garden is 200 square feet. So, we can write the equation:
2.5x + 8y = 200
Now, we can find three possible combinations by assigning some values to x and solving for y, while making sure that both x and y are positive integers.
Combination 1:
Let's assume x = 10 (10 small plants) and solve for y.
2.5(10) + 8y = 200
25 + 8y = 200
8y = 200 - 25
8y = 175
y = 175 / 8 ≈ 21.875
Since y cannot be a fraction, this combination is not valid.
Combination 2:
Let's assume x = 20 (20 small plants) and solve for y.
2.5(20) + 8y = 200
50 + 8y = 200
8y = 200 - 50
8y = 150
y = 150 / 8 ≈ 18.75
Since y cannot be a fraction, this combination is not valid either.
Combination 3:
Let's assume x = 30 (30 small plants) and solve for y.
2.5(30) + 8y = 200
75 + 8y = 200
8y = 200 - 75
8y = 125
y = 125 / 8 ≈ 15.625
Since y cannot be a fraction, this combination is also not valid.
None of the assumed values of x have given us valid combinations of small and large plants in the garden. It seems that there are no whole number solutions for this particular scenario.