The number,n (in hundreds), of mosquitoes in a camping area after t weeks can be modelled by the equation n(t)=2t^4-5t^3-16t^2+45t
According to this model, when will the population of mosquitoes be greater than 1800?
To find when the population of mosquitoes will be greater than 1800, we need to solve the equation n(t) > 1800.
The given equation is: n(t) = 2t^4 - 5t^3 - 16t^2 + 45t
So, we have:
2t^4 - 5t^3 - 16t^2 + 45t > 1800
To solve this inequality, we'll rearrange it to form a quadratic equation equal to zero:
2t^4 - 5t^3 - 16t^2 + 45t - 1800 > 0
Now, there are multiple ways to solve this inequality. One approach is as follows:
Step 1: Factor out common terms, if possible.
2t^4 - 5t^3 - 16t^2 + 45t - 1800 = (t - 4)(2t - 15)(t^2 + 10t + 60) > 0
Step 2: Find the critical points by setting each factor equal to zero and solving for t.
t - 4 = 0 => t = 4
2t - 15 = 0 => t = 15/2 = 7.5
t^2 + 10t + 60 = 0 => this quadratic factor does not have any real solutions, as its discriminant is negative.
Step 3: Plot these critical points on a number line.
We have t = 4 and t = 7.5 as critical points.
Step 4: Test intervals between critical points to determine the sign of the inequality.
Choose test points from each interval and substitute them into the inequality. For instance, you can use t = 0, t = 5, and t = 10.
Testing the interval t < 4:
When t = 0, (t - 4)(2t - 15)(t^2 + 10t + 60) = (-4)(-15)(60) = -36,000
Since the result is negative, this interval does not satisfy the inequality.
Testing the interval 4 < t < 7.5:
When t = 5, (t - 4)(2t - 15)(t^2 + 10t + 60) = (1)(-5)(235) = -1,175
Since the result is negative, this interval does not satisfy the inequality.
Testing the interval t > 7.5:
When t = 10, (t - 4)(2t - 15)(t^2 + 10t + 60) = (6)(5)(460) = 13,800
Since the result is positive, this interval satisfies the inequality.
Step 5: Determine the solution.
From the analysis above, we can conclude that the inequality is satisfied for t > 7.5. Therefore, the population of mosquitoes will be greater than 1800 when t is greater than 7.5 (weeks).
Note: This is an approximate solution based on the mathematical model. Actual conditions may vary.