There seems to be a strange typo in problem 1, but I'm not sure. So I'll skip it and move onto problem 2
The answer to problem 2 is y = -0.125x^2 + 5.25x - 35.05
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Explanation:
We have the following points:
(12, 9.95) and (14, 13.95) and (16,16.95)
where x is the diameter in inches, y is the price in dollars
Use these three points to find the equation in the form y = ax^2 + bx + c
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Plug in (x,y) = (12,9.95). Isolate c
y = ax^2 + bx + c
9.95 = a(12)^2 + b(12) + c
9.95 = 144a + 12b + c
9.95-144a-12b = c
c = 9.95-144a-12b
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Plug in (x,y) = (14, 13.95)
Also, plug in c = 9.95-144a-12b
y = ax^2 + bx + c
13.95 = a(14)^2 + b(14) + c
13.95 = 196a + 14b + c
13.95 = 196a + 14b + 9.95-144a-12b
13.95 = 52a + 2b + 9.95
Solve for b
13.95 = 52a + 2b + 9.95
13.95-9.95 = 52a + 2b
4 = 52a + 2b
2b = -52a + 4
b = (-52a + 4)/2
b = -26a + 2
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Plug in (x,y) = (16,16.95)
y = ax^2 + bx + c
16.95 = a(16)^2 + b(16) + c
16.95 = 256a + 16b + c
Now plug in c = 9.95-144a-12b
16.95 = 256a + 16b + c
16.95 = 256a + 16b + 9.95-144a-12b
16.95 = 112a + 4b + 9.95
16.95-9.95 = 112a + 4b
7 = 112a + 4b
112a + 4b = 7
Next, plug in b = -26a+2, and solve for 'a'
112a + 4b = 7
112a + 4(-26a+2) = 7
112a + 4(-26a)+4(2) = 7
112a - 104a + 8 = 7
8a + 8 = 7
8a = 7-8
8a = -1
a = -1/8
a = -0.125
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Use that 'a' value to find b
b = -26a + 2
b = -26(-0.125) + 2
b = 5.25
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Finally, use those 'a' and 'b' values to determine c
c = 9.95-144a-12b
c = 9.95-144(-0.125)-12(5.25)
c = -35.05
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We found that
a = -0.125, b = 5.25, and c = -35.05
So y = ax^2 + bx + c turns into y = -0.125x^2 + 5.25x - 35.05
As a way to check the answer, we plug in the various x values to get the corresponding y values. For example, if x = 12, then
y = -0.125x^2 + 5.25x - 35.05
y = -0.125(12)^2 + 5.25(12) - 35.05
y = 9.95
Which is the proper y value we want. I'll let you check the others
