The Golden Gate Bridge in San Francisco is constantly being repainted. It is estimated that the surface area of exposed steel that needs to be painted is approximately 10 million square feet. (i) Given that 1 metre is equal to 3.28 feet, convert the surface area of the bridge into square metres, giving your answer in the form a × 10ⁿ, where 1 ≤ a < 10 and n ∈ ℕ. Give the value of a correct to two significant figures. (ii) A litre of the paint used on the Golden Gate Bridge will cover approximately 5 square metres. This paint comes in 25 litre tins. Find the minimum number of tins of paint that will be needed to paint the entire bridge. (b) Solve the equation $2^{2x-1} = 8^{2x}$.

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The Golden Gate Bridge in San Francisco is constantly being repainted - Leaving Cert Mathematics - Question 4 - 2020

Question 4

The Golden Gate Bridge in San Francisco is constantly being repainted. It is estimated that the surface area of exposed steel that needs to be painted is approximate... show full transcript

Worked Solution & Example Answer:The Golden Gate Bridge in San Francisco is constantly being repainted - Leaving Cert Mathematics - Question 4 - 2020

Step 1

Convert the surface area to square metres

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Answer

To convert 10 million square feet to square metres, we first convert 10 million to its numerical value:

$10,000,000 ext{ square feet}$

Using the conversion factor, where 1 metre = 3.28 feet, we find:

$10000,000 ext{ square feet} = \frac{10,000,000}{(3.28)^2} ext{ square metres}$ $= \frac{10,000,000}{10.7584} ext{ square metres}$ $\approx 930,000 ext{ square metres}$

This can be expressed in scientific notation as:

$9.3 \times 10^5 ext{ square metres}$

Step 2

Calculate the number of tins of paint needed

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With the area needing paint calculated as 930,000 square metres and knowing that a litre of paint covers 5 square metres, we find the total litres required:

$\frac{930,000 ext{ square metres}}{5 ext{ square metres/litre}} = 186,000 ext{ litres}$

Since paint comes in 25 litre tins, we calculate the number of tins required:

$\frac{186,000 ext{ litres}}{25 ext{ litres/tin}} = 7,440 ext{ tins}$

Step 3

Solve the equation $2^{2x-1} = 8^{2x}$

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Answer

Starting with the equation:

$2^{2x-1} = 8^{2x}$

We can rewrite 8 as a power of 2:

$8 = 2^3$ Thus, we have:

$8^{2x} = (2^3)^{2x} = 2^{6x}$

Now our equation becomes:

$2^{2x-1} = 2^{6x}$

Since the bases are the same, we can equate the exponents:

$2x - 1 = 6x$

Solving for x gives us:

$-1 = 6x - 2x$ $-1 = 4x$ $x = -\frac{1}{4}$

The Golden Gate Bridge in San Francisco is constantly being repainted - Leaving Cert Mathematics - Question 4 - 2020

Worked Solution & Example Answer:The Golden Gate Bridge in San Francisco is constantly being repainted - Leaving Cert Mathematics - Question 4 - 2020

Convert the surface area to square metres

Calculate the number of tins of paint needed

Solve the equation $2^{2x-1} = 8^{2x}$

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