Need help with moderately hard A-level maths question

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Keifal246
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The cost of building a lighthouse is proportional to the cube of its height, h
The distance d that the top of the lighthouse can be seen from a point at sea level is modelled by d =√2RH, where R is the radius of the earth and d, R and h are in the same units
Three possible design x y and z are considered in which the top of the lighthouse can be seen at 20km, 40km and 60km, respectively.
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Keifal246
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Find the ratios of the costs of designs X,Y and Z.
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tiny hobbit
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(Original post by Keifal246)
Find the ratios of the costs of designs X,Y and Z.
Should the square root be over the R and H as well as the 2?

Either way, I would rearrange the formula for d to get h on its own. You can then write down the cost as k x h^3 and so in terms of d. Putting in the different d values gives you a ration from which the 2, k and R can be cancelled out, leaving a ratio just with numbers in.
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Deranging
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Is your formula supposed to be  d = \sqrt{2 R h} ?

To get you started, consider the related question, suppose I want to double the distance from which the top of the lighthouse can be seen, how much higher do I need to build the lighthouse? And thus, how much more does it cost?
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Keifal246
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(Original post by tiny hobbit)
Should the square root be over the R and H as well as the 2?

Either way, I would rearrange the formula for d to get h on its own. You can then write down the cost as k x h^3 and so in terms of d. Putting in the different d values gives you a ration from which the 2, k and R can be cancelled out, leaving a ratio just with numbers in.
Yes it should be around the 2Rh
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tiny hobbit
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(Original post by Keifal246)
Yes it should be around the 2Rh
So I should start by squaring both sides and then do the process I described above.
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MXNIXC
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Lmao, I have the same problem too. It's question 13 from the A levels Edexcel mathematics for year 1 and AS by porkess and Berry. Gonna ask my maths teacher.
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Keifal246
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(Original post by tiny hobbit)
So I should start by squaring both sides and then do the process I described above.
I've uploaded what I've done so far. How do I progress after that?
Attached files
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Akh40
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(Original post by Keifal246)
I've uploaded what I've done so far. How do I progress after that?
well you have the three costs and they all have a common factor. if you divide them all by K then multiply by R^3 you should get the following?
8000000:512000000:5832000000
divide all by a million
8:512:5832
divide all by 8
1:64:729
this could be even more simplified if you find the cube root
1:4:9
which again simplifies by finding the square root
1:2:3

which I believe is the answer, if your costs are correct
correct me if I am wrong please
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SerBronn
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You're really over-thinking this ...

The distances are in the ratio 1 : 2 : 3. If distances and heights were linearly related then the heights would be in this ratio too, but we are told that the distances are actually proportional to the square roots of the heights. So what ratio must the heights be in to give a ratio 1 : 2 : 3 in distances?
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Keifal246
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(Original post by Akh40)
well you have the three costs and they all have a common factor. if you divide them all by K then multiply by R^3 you should get the following?
8000000:512000000:5832000000
divide all by a million
8:512:5832
divide all by 8
1:64:729
this could be even more simplified if you find the cube root
1:4:9
which again simplifies by finding the square root
1:2:3

which I believe is the answer, if your costs are correct
correct me if I am wrong please
I believe u are wrong as ratios cannot be simplified by rooting. They have to be simplified using a factor that they share.
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Akh40
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ahh in which case it'd be 1:64:729
although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓
(Original post by Keifal246)
I believe u are wrong as ratios cannot be simplified by rooting. They have to be simplified using a factor that they share.
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Keifal246
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(Original post by Akh40)
ahh in which case it'd be 1:64:729
although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓
However, u agreed with yourself instead of correcting me so...
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Akh40
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(Original post by Keifal246)
However, u agreed with yourself instead of correcting me so...
yeah I agreed it seemed right which is why I mentioned it today too
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tiny hobbit
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(Original post by Keifal246)
I believe u are wrong as ratios cannot be simplified by rooting. They have to be simplified using a factor that they share.
(Original post by Akh40)
ahh in which case it'd be 1:64:729
although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓
Correct, you can't simplify by rooting, so stick with 1:64:729 i.e. 1, 2 and 3 each to the power 6.
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