Answer:
The x-intercept represents that after 10 in. of rain there are mosquitoes.
Step-by-step explanation:
The population of mosquitoes in a certain area as a function of inches of rainfall is modeled in the table.
The function is quadratic, [tex]P(x)=ax^2+bx+c[/tex]
where, P represents number of mosquitoes in millions. x represents rainfall in inches.
Table :
x : 0 1 3 4 5 6
P : 0 9 21 24 25 24
Using three point to make system of equation and solve for coefficients a, b and c
Put (0,0) and we get [tex]c=0[/tex]
Put (1,9) and we get [tex]a+b+c=9[/tex]
Put (3,21) and we get [tex]9a+3b+c=21[/tex]
Solve for a, b and c
[tex]a=-1[/tex]
[tex]b=10[/tex]
[tex]c=0[/tex]
[tex]P(x)=-x^2+10x[/tex]
Now, we will find x-intercept of the function. Put P(x)=0 and solve for x
[tex]-x^2+10x=0[/tex]
[tex]x=0,10[/tex]
The x-intercepts represent number of mosquitoes after 10 in of rain fall.
The x-intercept represents that after 10 in. of rain there are mosquitoes, when the population of mosquitoes in a certain area as a function of inches of rainfall.
A quadratic equation is the equation in which the unknown variable is one and the highest power of the unknown variable is two.
The population of mosquitoes in a certain area as a function of inches of rainfall is modeled in the table.
The function is quadratic. The standard form of the quadratic equation is,
[tex]f(x)=ax^2+bx+c[/tex]
Here (a,b, c) is the real numbers and (x) is the variable. For the first point (0,0). The equation become,
[tex]f(0)=a(0)^2+b(0)+c\\0=c[/tex]
For the second point (1,9). The equation become,
[tex]f(1)=a(1)^2+b(1)+c\\9=a+b+0\\a+b=9[/tex] ...1
For the third point (1,9). The equation become,
[tex]f(3)=a(3)^2+b(3)+c\\21=9a+3b+0\\9a+3b=21[/tex] ...2
On solving, first and second equation we get,
[tex]a=-1\\b=10[/tex]
Put the values of a,b, and c in a quadratic equation,
[tex]f(x)=-x^2+10x+0[/tex]
Equate the equation to zero,
[tex]-x^2+10x=0\\x(x-10)=0\\x=0, x=10[/tex]
Thus, the x-intercept represents that after 10 in. of rain there are mosquitoes, when the population of mosquitoes in a certain area as a function of inches of rainfall.
Learn more about the quadratic equation here;
https://brainly.com/question/1214333
#SPJ3
