1. **Write the proportionality statement as an equation.**
d is directly proportional to the square of t, so d ∝ t².
This can be written as d = kt², where k is the constant of proportionality.
2. **Use the given information to find the value of k.**
We are told d = 20 when t = 2.
Substitute these values into the equation:
20 = k * (2)²
20 = k * 4
k = 20 / 4
k = 5
3. **Write the final equation.**
Now we know k, the equation is d = 5t².
4. **Use the equation to find the time to reach the ground.**
The stone reaches the ground when it has fallen 300 metres, so d = 300.
300 = 5t²
Divide by 5:
t² = 300 / 5
t² = 60
Take the square root:
t = √60
t ≈ 7.7459...
The total time taken is approximately 7.75 seconds (to 3 s.f.).
The problem involves direct proportion.
Step 1: Set up the proportionality equation.
"d is directly proportional to the square of t" means d ∝ t².
We can write this as an equation with a constant of proportionality, k:
d = kt²
Step 2: Find the constant k.
We are given that the stone falls 20 metres (d=20) in 2 seconds (t=2). We substitute these values into our equation to find k.
20 = k * (2)²
20 = k * 4
k = 20 / 4 = 5
Step 3: Write the specific formula.
Now that we know k=5, our formula is:
d = 5t²
Step 4: Solve for the required time.
We want to find the time it takes for the stone to reach the ground, which is a fall of 300 metres (d=300).
We use our formula and solve for t:
300 = 5t²
Divide both sides by 5:
t² = 300 / 5
t² = 60
Take the square root of both sides:
t = √60 ≈ 7.746 seconds.
Rounding to a suitable degree of accuracy, for example, 3 significant figures, gives 7.75 seconds.
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