darla designed a stained glass window that incorporates 20 squares, triangles, and circles altoghether. the window has 2 more triangles than twice the number of squares. the number of circles in the window is a multiple of three. how many of each shape appears in darla's window if there are at least two of each shape? please help me asap it is due tomrrow
squares = 5
triangles = 12
circles = 3
???
To solve this problem, let's represent the number of squares, triangles, and circles as variables: let S be the number of squares, T be the number of triangles, and C be the number of circles.
From the given information, we can create three equations:
Equation 1: The total number of shapes is 20:
S + T + C = 20
Equation 2: There are 2 more triangles than twice the number of squares:
T = 2S + 2
Equation 3: The number of circles is a multiple of three:
C = 3N (where N is a whole number)
We are also given that there are at least two of each shape, so S, T, and C are all greater than or equal to 2.
To solve this system of equations, we can use substitution or elimination method. Let's use substitution:
1. Substitute equation 2 into equation 1:
S + (2S + 2) + C = 20
3S + 2 + C = 20
3S + C = 18
2. Substitute equation 3 into equation 1:
S + T + (3N) = 20
S + T + 3N = 20
Now we have two equations with two variables:
3S + C = 18 (Equation A)
S + T + 3N = 20 (Equation B)
To find the values of S, T, and C, we can try different values for N and check if there is a combination that satisfies both equations. Since there are at least two of each shape, let's start with N = 2:
Substituting N = 2 into equation B:
S + T + 3(2) = 20
S + T + 6 = 20
S + T = 14
Substituting N = 2 and the value of S + T into equation A:
3S + C = 18
3(14 - T) + C = 18
42 - 3T + C = 18
C = 3T - 24
Now let's try different values of T, while keeping in mind the constraints that S, T, and C are greater than or equal to 2.
When T = 2, C = 3(2) - 24 = -18 (which violates the constraints)
When T = 3, C = 3(3) - 24 = -15 (which violates the constraints)
When T = 4, C = 3(4) - 24 = -12 (which violates the constraints)
When T = 5, C = 3(5) - 24 = -9 (which violates the constraints)
When T = 6, C = 3(6) - 24 = -6 (which violates the constraints)
When T = 7, C = 3(7) - 24 = -3 (which violates the constraints)
When T = 8, C = 3(8) - 24 = 0 (which satisfies the constraints)
Now we have the values of S, T, and C:
S = 14 - T = 14 - 8 = 6
T = 8
C = 0
Therefore, in Darla's window, there are 6 squares, 8 triangles, and 0 circles.