# Need help with moderately hard A-level maths question

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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.

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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#3

(Original post by

Find the ratios of the costs of designs X,Y and Z.

**Keifal246**)Find the ratios of the costs of designs X,Y and Z.

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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#4

Is your formula supposed to be ?

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?

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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(Original post by

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.

**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.

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#6

(Original post by

Yes it should be around the 2Rh

**Keifal246**)Yes it should be around the 2Rh

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#7

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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(Original post by

So I should start by squaring both sides and then do the process I described above.

**tiny hobbit**)So I should start by squaring both sides and then do the process I described above.

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#9

(Original post by

I've uploaded what I've done so far. How do I progress after that?

**Keifal246**)I've uploaded what I've done so far. How do I progress after that?

80000005832000000

divide all by a million

85832

divide all by 8

1729

this could be even more simplified if you find the cube root

19

which again simplifies by finding the square root

13

which I believe is the answer, if your costs are correct

correct me if I am wrong please

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#10

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?

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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(Original post by

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?

80000005832000000

divide all by a million

85832

divide all by 8

1729

this could be even more simplified if you find the cube root

19

which again simplifies by finding the square root

13

which I believe is the answer, if your costs are correct

correct me if I am wrong please

**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?

80000005832000000

divide all by a million

85832

divide all by 8

1729

this could be even more simplified if you find the cube root

19

which again simplifies by finding the square root

13

which I believe is the answer, if your costs are correct

correct me if I am wrong please

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#12

ahh in which case it'd be 1729

although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓

although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓

(Original post by

I believe u are wrong as ratios cannot be simplified by rooting. They have to be simplified using a factor that they share.

**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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(Original post by

ahh in which case it'd be 1729

although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓

**Akh40**)ahh in which case it'd be 1729

although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓

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#14

(Original post by

However, u agreed with yourself instead of correcting me so...

**Keifal246**)However, u agreed with yourself instead of correcting me so...

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#15

**Keifal246**)

I believe u are wrong as ratios cannot be simplified by rooting. They have to be simplified using a factor that they share.

**Akh40**)

ahh in which case it'd be 1729

although it was you who suggested we simplify by finding the root yesterday after school .. 🤓🤓

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