1. **Find the value of a.**
The constant 'a' represents the initial value, which is the value of d when t = 0.
From the graph, when t = 0, the curve intersects the d-axis at d = 5.
So, **a = 5**.
(Mathematically: d = a × b⁰ => d = a × 1 => d = a)
2. **Find the value of b.**
Now we know the equation is d = 5 × bᵗ.
We can pick another point from the graph to find b. Let's choose an easy point to read, like (t=2, d=20).
Substitute these values into the equation:
20 = 5 × b²
Divide by 5:
4 = b²
Take the square root:
b = 2 (we take the positive root as the graph is increasing).
So, **b = 2**.
3. **Check with another point.**
Let's check with (t=3, d=40).
d = 5 × 2³ = 5 × 8 = 40. This matches the graph, so our values are correct.
Final answer: a = 5, b = 2.
The equation given is for an exponential curve: d = a × bᵗ.
Step 1: Find the value of `a`.
The constant `a` is the initial value, or the y-intercept, where t=0.
Let's substitute t=0 into the equation:
d = a × b⁰
Since any number raised to the power of 0 is 1 (b⁰ = 1), the equation becomes:
d = a × 1
d = a
So, `a` is the value of `d` when t=0. Looking at the graph, the curve crosses the d-axis at 5.
Therefore, **a = 5**.
Step 2: Find the value of `b`.
Now we know the equation is d = 5 × bᵗ. We can find `b` by substituting the coordinates of another point from the curve. Let's choose a point that is easy to read accurately, for example, the point (t=2, d=20).
Substitute t=2 and d=20 into the equation:
20 = 5 × b²
To solve for b, first divide both sides by 5:
20 / 5 = b²
4 = b²
Now, take the square root of both sides:
b = √4
b = 2
(We take the positive root because the graph shows growth).
Therefore, **b = 2**.
The final equation is d = 5 × 2ᵗ. We can check this with another point, (t=3, d=40): 40 = 5 × 2³, which is 40 = 5 × 8. This is correct.
Expert