An archer shoots an arrow - Edexcel - A-Level Maths Pure - Question 12 - 2017 - Paper 1
Question 12
An archer shoots an arrow.
The height, H metres, of the arrow above the ground is modelled by the formula
$$H = 1.8 + 0.4d - 0.002d^{2}, \, d > 0$$
where d is the... show full transcript
Worked Solution & Example Answer:An archer shoots an arrow - Edexcel - A-Level Maths Pure - Question 12 - 2017 - Paper 1
Step 1
Find the horizontal distance travelled by the arrow, as given by this model.
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Answer
To find the horizontal distance where the arrow hits the ground, set the height H to 0:
0=1.8+0.4d−0.002d2
Rearranging gives:
0.002d2−0.4d−1.8=0
Using the quadratic formula, where a = 0.002, b = -0.4, and c = -1.8:
d=2a−b±b2−4ac
Calculating the discriminant:
b2−4ac=(−0.4)2−4(0.002)(−1.8)=0.16+0.0144=0.1744
Now solving for d:
d=0.0040.4±0.1744=0.0040.4±0.4177
This provides two solutions, but we only consider the positive:
Step 2
With reference to the model, interpret the significance of the constant 1.8 in the formula.
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Answer
The constant 1.8 in the formula represents the initial height of the arrow above the ground when the horizontal distance d is zero. This means that when the arrow is shot, it starts at a height of 1.8 metres off the ground.
Step 3
Write 1.8 + 0.4d - 0.002d^{2} in the form A - B(d - C)^{2}
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Answer
To rewrite the equation in the desired form, we first complete the square:
Starting with:
H=1.8+0.4d−0.002d2
Factor out -0.002:
H=−0.002(d2−200d)+1.8
Next, complete the square for d2−200d:
=−0.002((d−100)2−10000)+1.8
Distributing -0.002:
H=−0.002(d−100)2+(1.8+20)=−0.002(d−100)2+21.8
Thus, we have:
A=21.8,B=0.002,C=100.
Step 4
the maximum height of the arrow above the ground.
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Answer
For the adapted model, the maximum height can be found by evaluating at the vertex of the parabola described by the formula:
H=2.1+0.4d−0.002d2
The vertex occurs at:
d=−2ab=−2(−0.002)0.4=100extm
Evaluating H at d = 100:
H=2.1+0.4(100)−0.002(100)2
Substituting:
H=2.1+40−200=42.1extm
Step 5
the horizontal distance, from the archer, of the arrow when it is at its maximum height.
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Answer
The horizontal distance when the arrow reaches its maximum height is already calculated as:
