A function h(t) increases by a factor of 10 over every unit interval in t and h(0)=1.Which could be a function rule for h(t)?

To determine a function rule for $h(t)$ that increases by a factor of 10 over every unit interval, we can analyze the problem step by step.

1. ### Break Down the Problem

• We know that $h(0) = 1$.
• The function increases by a factor of 10 for every unit increase in $t$. This means:
  • $h(1) = 10 \cdot h(0)$
  • $h(2) = 10 \cdot h(1)$
• Therefore, we need a function that captures this exponential growth.

2. ### Relevant Concepts

The general form of an exponential growth function can be expressed as: $h(t) = h(0) \cdot r^t$ where $r$ is the growth factor.

3. ### Analysis and Detail

Given $h(0) = 1$ and the growth factor is 10, we can substitute these values into the formula: $h(t) = 1 \cdot 10^t = 10^t$

4. ### Verify and Summarize

Let's verify:

• $h(0) = 10^0 = 1$
• $h(1) = 10^1 = 10$
• $h(2) = 10^2 = 100$ The function increases by a factor of 10 for each increment of 1 in $t$, confirming that $h(t) = 10^t$ is correct.

Final Answer

The function rule for $h(t)$ is: $h(t) = 10^t$