The function is: f ( x ) = - x² + 60 x - 116 is in the form: f ( x ) = a x² + b x + c
Part A:
x = -b / 2 a = - 60 / ( - 2 ) = 30
Then we will charge it into the function:
f ( 30 ) = - 30² + 60 · 30 - 116 = - 900 +1,800 - 116 = 784
The vertex is ( 30, 784 ).
It means that the maximum profit is $784, when 30 windshields are repaired.
Part B:
x 1/2 = ( -60 +/- √(3600 - 464 )/ (- 2 ) = ( -60 +/- 56 ) / ( - 2 )
x 1 = 2, x 2 = 58
It means that a shop has to repair at least 2 windshields and not more than 58 windshields for making a profit.
The required vertex of the parabola is (30,784) and x-intercept is x = 2
A parabola is a cross-section cut out of the cone and represented by an equation [tex]y =4ax^2[/tex].
Part A:
and x = 58.The standard form of the parabola [tex]4p\left(y-k\right)=\left(x-h\right)^2[/tex].
given equation of parabola is f(x) = –x² + 60x – 116 reduce this equation to standard form, f(x) = y
y = –x² + 60x – 116
y = -(x - 30)² + 784
y - 784 = - (x - 30)²
4(1/4)(y - 784) = -(x - 30)² - - - - - - (1)
From the above equation vertex ( h, k ) = (30 , 784)
Part B:
Now, x-intercept, put y = 0 in equation 1
(x - 30)² = 784
x - 30 = ± 28
x = 30 ± 28
x = 2 and x = 58 is the required x-intercepts.
Thus , the required vertex of the parabola is (30, 784) and x-intercept is x = 2 and x = 58.
Learn more about parabola here:
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