Write an exponential equation in the form y = a(b)^x that can model the number of views, y, x days after Jack first watched the video. To the nearest hundred views, how many views... Write an exponential equation in the form y = a(b)^x that can model the number of views, y, x days after Jack first watched the video. To the nearest hundred views, how many views can Jack expect the video to have 7 days after he first watched it? Read more

Steps to Solve

1. Identify Initial Value and Growth Factor

The initial number of views when Jack first watched the video is $ a = 3140 $ (at $ x = 0 $). The number of views after one day (when $ x = 1 $) is $ y = 5024 $.

2. Set Up the Equation

We will use the exponential growth model:

$$ y = a(b)^x $$

Substituting the known values:

$$ 5024 = 3140(b)^1 $$

3. Solve for Growth Factor (b)

To find $ b $, rearrange the equation:

$$ b = \frac{5024}{3140} $$

Calculating:

$$ b \approx 1.60 $$

4. Write the Exponential Equation

Now substituting $ a $ and $ b $ back into the equation:

$$ y = 3140(1.6)^x $$

5. Predict Views After 7 Days

Now we need to find views after 7 days, so substitute $ x = 7 $ into the equation:

$$ y = 3140(1.6)^7 $$

Calculating $ (1.6)^7 $ yields approximately $ 18.78 $:

$$ y \approx 3140 \times 18.78 $$

6. Calculate Final Number of Views

Calculating the product gives:

$$ y \approx 59090.8 $$

Rounding to the nearest hundred:

$$ y \approx 59100 $$