Question

The population of mosquitoes in a certain area as a function of inches of rainfall is modeled in the table. The function is quadratic,
RAINFALL(in.) .POPULATION (millions.)
0. . 0
1. . 9
3. . 21
4. .24
5. . 25
6. . 24

The x-intercept represents that after _____ in. of rain there are mosquities.
( I cant seem to wrap my head around this, If you can just help me start somehwere i'd really appriciate it. Also I noticed the pattern in mosquitoes.)

Answer 1

Answer:

The x-intercept represents that after 10  in. of rain there are mosquitoes.

                        Or

The x-intercept represents that after 0 in. of rain there are no mosquitoes.

Step-by-step explanation:

We are given a table that represents the population of mosquitoes in a certain area as a function of inches of rainfall :

RAINFALL(in.)     POPULATION (millions.)

     0                               0

      1                                9

      3                              21

      4                              24

      5                              25

      6                              24

We will first find the function P(t) which denotes the number of mosquitoes after t inches of rainfall.

As the function is quadratic.

Hence , let:

$P(t)=at^2+bt+c$

At t=0 we have, P(t)=0

$c=0$

at t=1 we have: P(t)=9

i.e.

$a+b=9-----(1)$

Also at t=2 we have: P(t)=21

i.e.

$9a+3b=21-------(2)$

On solving equation (1) and (2) we get:

a= -1 and b=10

Hence,

$P(t)=-t^2+10t$

We know that the x-intercept of a graph is a point where the y-value is zero.

i.e. it is the inches of rainfall when the population of mosquitoes is zero i.e. there are no mosquitoes.

i.e. we have to find t such that:

P(t)=0

i.e.

$-t^2+10t=0\\\\i.e.\\\\t(10-t)=0\\\\i.e.\\\\ t=0\ or\ t=10$

Hence, we have got two x-intercept.

one at t=0 and the other at t=10.

The x-intercept represents that after 10 in. of rain there are no mosquitoes.

         Or

The x-intercept represents that after 0 in. of rain there are no mosquitoes.

Answer 2

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, $P(x)=ax^2+bx+c$

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  $c=0$

Put (1,9) and we get $a+b+c=9$

Put (3,21) and we get $9a+3b+c=21$

Solve for a, b and c

$a=-1$

$b=10$

$c=0$

$P(x)=-x^2+10x$

Now, we will find x-intercept of the function. Put P(x)=0 and solve for x

$-x^2+10x=0$

$x=0,10$

The x-intercepts represent number of mosquitoes after 10 in of rain fall.