Answer: 15 squash, 5 pumpkins; 15 squash, 10 pumpkins; 20 squash, 5 pumpkins.
Step-by-step explanation:
Then graph the system of inequalities: 3.5x + 5.59y (less than or equal to) 117; x + y > 10.
The first inequality is saying the cost of x squash and y pumpkins should total no more than $117.
The second inequality states x squash and y pumpkins should add up to more than 10.
X + k y = 1
k x + y = 1 / * ( - k )
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x + k y = 1
- k² x - k y = - k
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x - k² x = 1 - k
x ( 1 - k² ) = 1 - k
x = ( 1 - k ) / ( 1 - k² ) = ( 1 - k ) / ( 1 - k ) ( 1 + k )
y = 1 - k( 1 - k )/( 1 - k² )
y = ( 1 - k ) / ( 1 - k² ) = ( 1 - k ) / ( 1 - k ) ( 1 + k )
a ) For k = - 1 this system has no solution.
b ) For k ≠ - 1 and k ≠ 1, the system has unique solution:
( x , y ) = ( 1/ (1 + k) , 1/( 1 + k ) ).
c ) For k = 1, there are infinitely many solutions.
S casts a "shadow" on the x-y plane given by the set
Each cross section is a square whose side length is determined by the vertical distance between y = 1 and y = x², which is |1 - x²|. But since -1 ≤ x ≤ 1, this distance simplifies to 1 - x².
The volume of an infinitesimally thin section is then (1 - x²)² ∆x (where ∆x represents its thickness), and so the volume of S is
The integrand is even, so this integral is equal to twice the integral over [0, 1] :
