Answer:
8 inches of rainfall will maximize the number of mosquitoes.
Maximum number of mosquitoes = 64 millions
Step-by-step explanation:
We are given the following in the question:
[tex]M(x) =16x-x^2[/tex]
where M(x) is the number of mosquitoes in millions and x is the rainfall in inches.
First, we differentiate M(x) with respect to x, to get,
[tex]\dfrac{d(M(x))}{dx} = \dfrac{d(16x-x^2)}{dx} = 16-2x[/tex]
Equating the first derivative to zero, we get,
[tex]\dfrac{d(M(x))}{dx} = 0\\\\16-2x = 0[/tex]
[tex]x =8[/tex]
Again differentiation M(x), with respect to x, we get,
[tex]\dfrac{d^2(M(x))}{dx^2} = -2[/tex]
[tex]\dfrac{d^2(M(x))}{dx^2} < 0[/tex]
Thus, by double differentiation test the maxima occurs at x = 8 for M(x).
Thus, maximum number of mosquitoes are there when the rainfall is 8 inches.
Maximum number of mosquito:
[tex]M(8) =16(8)-(8)^2 = 64[/tex]
Thus, maximum number of mosquito is 64 millions.
Vertex: (8,64)
The attached image shows the graph.
