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A planet takes $T$ days to complete one orbit of the Sun - AQA - A-Level Maths: Pure - Question 7 - 2022 - Paper 3
Question 7
A planet takes $T$ days to complete one orbit of the Sun. $T$ is known to be related to the planet's average distance $d$, in millions of kilometres, from the Sun. A... show full transcript
Worked Solution & Example Answer:A planet takes $T$ days to complete one orbit of the Sun - AQA - A-Level Maths: Pure - Question 7 - 2022 - Paper 3
Step 1
Find the equation of the straight line in the form $ ext{log}_{10} T = a + b ext{log}_{10} d$
Answer
To find the equation of the straight line, we will use the coordinates of the two known points from the graph:
- Mercury:
- Uranus:
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Calculate the slope (b) using the formula: b = rac{y_2 - y_1}{x_2 - x_1} = rac{4.49 - 1.94}{3.46 - 1.76} = rac{2.55}{1.70} = 1.5
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Use point-slope form to find a: Select one point, let's use the coordinates for Mercury (1.76, 1.94). The equation becomes:
Now, rearranging to find gives:
- Final equation: The equation of the line is:
Step 2
Show that $T = K d^n$ where K and n are constants to be found.
Answer
Using the equation we just derived:
We can convert this equation from logarithmic form to exponential form. Using the property of logarithms:
Thus, we can express in terms of :
- Rearranging the equation:
- Exponentiate both sides:
- This simplifies to:
- We can define constants and as follows: Therefore, we have shown that:
Step 3
Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune from the Sun.
Answer
Given that Neptune takes approximately 60,000 days to complete one orbit of the Sun, we can substitute and into the equation:
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First, calculate : Using
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Substituting values into the equation:
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Rearranging to solve for d: