The distance of a particle from a point is d metres after t seconds. The relationship is given by d = a × bᵗ where a and b are constants. Use the graph to work out the values of a and b.

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.