SOLUTION: The number of tickets sold each day for an upcoming performance of Handel's Messiah is given by N(x)=-0.4x^2+8x+12, where x is the number of days since the concert was first announ

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Question 164164: The number of tickets sold each day for an upcoming performance of Handel's Messiah is given by N(x)=-0.4x^2+8x+12, where x is the number of days since the concert was first announced. When will daily ticket sales peak and how many tickets will be sold that day?
Answer by alicealc(293) About Me  (Show Source):
You can put this solution on YOUR website!
N(x)=-0.4x^2+8x+12

we can use differentiation for this problem to find the optimum value
the optimum value is reached when
dN%28x%29%2Fdx+=+0
-0.4*2*x + 8 = 0
<=> -0.8x + 8 = 0
<=> -0.8x = -8
<=> x = 8/0.8 = 10
because the coefficient of x^2 is negative, the function will have a maximum value.
the maximum value is:
N%2810%29+=+-0.4%2A10%5E2+%2B+8%2A10+%2B+12
N(10) = -0.4*100 + 80 + 12
= -40 + 80 + 12
= 52


or in another way, to find the maximum or minimum value of a quadratic function, this formula can be used:
because the coefficient of x^2 is negative, the function will have a maximum value.
the maximum value = D%2F%28-4a%29 -> D+=+b%5E2+-+4%2Aa%2Ac

from the function above, the value of a = -0.4, b = 8, c = 12
so the maximum value
= %288%5E2+-+4%2A%28-0.4%29%2A12%29%2F%28-4%2A%28-0.4%29%29
= %2864+%2B+19.2%29%2F1.6
= 83.2/1.6
= 52