Question

5.2: tiktok followers- ms. charlotte’s popularity on tiktok has increased. since january 2024, the number of subscribers she has has been increasing at an exponential rate. a). the number of subscribers s is a function of time, t, the number of months since january 2024. write an equation defining s. b). how many subscribers (¿cuántos suscriptores) did they have in january 2025? c). by what factor did the number of subscribers change: a). every 6 months? b). every year?

Explanation:

Part a)

Step 1: Identify the form of exponential function

The general form of an exponential function is \( S(t)=S_0 \cdot a^t \), where \( S_0 \) is the initial value, \( a \) is the growth factor, and \( t \) is time (in months). From the graph, when \( t = 0 \) (January 2024), \( S(0)=250000 \), so \( S_0 = 250000 \).

Step 2: Find the growth factor \( a \)

We know another point, say \( t = 1 \), \( S(1)=430000 \) (from the graph: \( (1, 430000) \)). Substitute into the exponential function: \( 430000=250000\cdot a^1 \). Solve for \( a \): \( a=\frac{430000}{250000}=\frac{43}{25} = 1.72 \).

Step 3: Write the equation

Substitute \( S_0 = 250000 \) and \( a = 1.72 \) into the general form: \( S(t)=250000\cdot(1.72)^t \).

Step 1: Determine the time \( t \)

January 2025 is 12 months after January 2024, so \( t = 12 \).

Step 2: Substitute into the equation

Use the equation from part (a): \( S(12)=250000\cdot(1.72)^{12} \). First, calculate \( (1.72)^{12} \). Let's compute step by step: \( 1.72^2 = 2.9584 \), \( 1.72^4=(2.9584)^2\approx8.7421 \), \( 1.72^8=(8.7421)^2\approx76.424 \), then \( 1.72^{12}=1.72^8\times1.72^4\approx76.424\times8.7421\approx668.0 \). Then \( S(12)=250000\times668.0 = 167000000 \) (approximate, more accurately using calculator: \( 1.72^{12}\approx1.72\times1.72 = 2.9584; 2.9584\times1.72\approx5.0884; 5.0884\times1.72\approx8.7521; 8.7521\times1.72\approx15.0536; 15.0536\times1.72\approx25.8922; 25.8922\times1.72\approx44.5346; 44.5346\times1.72\approx76.6005; 76.6005\times1.72\approx131.7529; 131.7529\times1.72\approx226.6149; 226.6149\times1.72\approx389.7776; 389.7776\times1.72\approx670.4175 \). Then \( S(12)=250000\times670.4175\approx167604375 \)).

Step 1: Identify the time interval

We need to find the factor by which the number of subscribers changes every 6 months. Let \( t = 0 \) and \( t = 6 \).

Step 2: Use the exponential function

From \( S(t)=250000\cdot(1.72)^t \), for \( t = 6 \), \( S(6)=250000\cdot(1.72)^6 \). Calculate \( (1.72)^6 \): \( 1.72^2 = 2.9584 \), \( 1.72^3=1.72\times2.9584\approx5.0884 \), \( 1.72^6=(5.0884)^2\approx25.892 \). So the factor is \( (1.72)^6\approx25.89 \) (or more accurately, \( 1.72^6 = 1.72\times1.72\times1.72\times1.72\times1.72\times1.72\approx25.89 \)). Alternatively, since the growth factor per month is \( 1.72 \), the growth factor per 6 months is \( (1.72)^6 \approx 25.9 \).

Answer:

\( S(t) = 250000 \cdot (1.72)^t \)

Part b)