A new song has gone viral on the Internet. The website hosting the song uses the function \large f\left(t\right)=200t^2 to represent the number of daily hits over time, where t is time in days. Use the function to predict the day on which the number of daily hits reaches 1,000,000. Show all work to receive credit for this question.
We want to find the value of t when f(t)=1000000. Substituting this into the equation and solving for t, we get:
1000000 = 200t^2
Dividing both sides by 200:
5000 = t^2
Taking the square root of both sides:
t = ±sqrt(5000)
Since time cannot be negative, we take the positive value:
t ≈ 70.71
Therefore, the number of daily hits will reach 1,000,000 on approximately the 71st day after the song went viral.
To predict the day on which the number of daily hits reaches 1,000,000, we can set the function f(t) equal to 1,000,000 and solve for t.
The given function is f(t) = 200t^2.
Setting it equal to 1,000,000, we have 200t^2 = 1,000,000.
Dividing both sides by 200, we get t^2 = 5,000.
Taking the square root of both sides, we have t = √(5,000).
Using a calculator, we find that √(5,000) is approximately 70.7107.
Therefore, the number of daily hits reaches 1,000,000 approximately on the 71st day.
To find the day on which the number of daily hits reaches 1,000,000, we need to solve the equation:
1,000,000 = 200t^2
First, divide both sides of the equation by 200 to isolate t^2:
t^2 = 1,000,000 / 200
t^2 = 5,000
Next, take the square root of both sides to solve for t:
t = √5,000
Now, let's simplify √5,000:
t ≈ √(2,500 * 2)
We can further simplify this expression:
t ≈ √2,500 * √2
The square root of 2,500 is 50:
t ≈ 50 * √2
So, t is approximately equal to 50 times the square root of 2.
Therefore, the day on which the number of daily hits reaches 1,000,000 is approximately 50√2 days.