Hi i need help on solving this question.
The population, P(t) of a city is modelled by the function P(t)=14t^2 + 650t + 32000
When will the population reach 50000?
I want to know what we are looking for, and how to get to that answer. Tnx
St see e
Dyes
To find out when the population reaches 50,000, we need to solve the equation P(t) = 50,000. The equation is given as P(t) = 14t^2 + 650t + 32,000.
To solve this equation, follow these steps:
Step 1: Set up the equation. Replace P(t) with 50,000.
50,000 = 14t^2 + 650t + 32,000
Step 2: Rearrange the equation. Move all terms to one side, creating a quadratic equation.
14t^2 + 650t + 32,000 - 50,000 = 0
Simplify the equation:
14t^2 + 650t - 18,000 = 0
Step 3: Solve the quadratic equation. There are multiple methods to do this, such as factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation does not factor easily, so we will use the quadratic formula.
The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 14, b = 650, and c = -18,000.
t = (-650 ± √(650^2 - 4 * 14 * -18,000)) / (2 * 14)
Step 4: Calculate the result using the quadratic formula.
t = (-650 ± √(422,500 + 1,008,000)) / 28
t = (-650 ± √(1,430,500)) / 28
t ≈ (-650 ± 1,196.03) / 28
Now we have two possible solutions for t:
t1 ≈ (-650 + 1,196.03) / 28
t2 ≈ (-650 - 1,196.03) / 28
Simplifying further:
t1 ≈ 546.03 / 28 ≈ 19.5
t2 ≈ -1,846.03 / 28 ≈ -65.9
Step 5: Interpret the results.
The population reaches 50,000 at two different times: approximately 19.5 and -65.9. However, time cannot be negative in this context, so we discard the negative value.
Therefore, the population will reach 50,000 at approximately t = 19.5.