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 inequality n(t) > 1800, using the given equation n(t) = 2t^4-5t^3-16t^2+45t.
Step 1: Write the inequality
2t^4-5t^3-16t^2+45t > 1800
Step 2: Rearrange the inequality
2t^4-5t^3-16t^2+45t - 1800 > 0
Step 3: Simplify the left side of the inequality
2t^4-5t^3-16t^2+45t - 1800 = 2t^4-5t^3-16t^2+45t - 1800
Step 4: Set the polynomial expression equal to zero by subtracting 1800 from both sides
2t^4-5t^3-16t^2+45t - 1800 = 0
Now, we need to solve this equation to find the values of t that satisfy the inequality.
Step 5: Solve the equation
Unfortunately, there is no direct algebraic way to solve a quartic equation (an equation with the highest power of t being 4). However, we can use numerical methods or graphing to estimate the values of t when the population of mosquitoes is greater than 1800.
You can use graphing software or online graphing tools to plot the graph of the equation n(t) = 2t^4-5t^3-16t^2+45t and see where it crosses the line y = 1800. The x-coordinate of the points of intersection will give you the values of t when the population of mosquitoes is greater than 1800.
Alternatively, you can use numerical methods like the bisection method or Newton's method to find the approximate values of t.
By applying these methods, you will be able to determine when the population of mosquitoes is greater than 1800 according to the given model.