The number of Mosquitoes M9x), in millions, in a certain area depends on the June rainfall x, in inches:M(x)=19x-x^2. What rainfall produces the maximum number of mosquitos?
Your function M(x) = 19x - x^2 is represented by a parabola which opens downwards
find the x of its vertex, that will be the rainfall which causes the most mosquitoes
To find the rainfall that produces the maximum number of mosquitoes, we can use calculus. We need to find the derivative of the mosquito population function, set it equal to 0, and solve for x.
1. Take the derivative of M(x) with respect to x:
M'(x) = 19 - 2x
2. Set the derivative equal to 0 and solve for x:
19 - 2x = 0
2x = 19
x = 9.5
Therefore, a rainfall of 9.5 inches produces the maximum number of mosquitoes.
To find the rainfall that produces the maximum number of mosquitoes, we need to determine the maximum value of the function M(x) = 19x - x^2.
One way to find the maximum is by taking the derivative of the function M(x) with respect to x and setting it equal to zero. The derivative will give us the slope of the function, and the point where the slope is zero represents a potential maximum.
So, let's find the derivative of M(x) using the power rule of differentiation.
M'(x) = d/dx(19x - x^2)
= 19 - 2x
Now, we set M'(x) equal to zero and solve for x:
19 - 2x = 0
2x = 19
x = 19/2
x = 9.5
Therefore, the rainfall that produces the maximum number of mosquitoes is 9.5 inches.