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 $a \times 10^n$, where $1 \leq a < 10$ and $n \in \mathbb{N}$. 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 $2^{x-1} = 8^{2x}$.

Leaving Cert Mathematics Indices: The Golden Gate Bridge in San Fr

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

Given that 1 metre is equal to 3.28 feet, convert the surface area of the bridge into square metres

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Answer

To convert the surface area from square feet to square metres, we first convert the area of 10 million square feet.

Calculate the conversion factor from square feet to square metres:

$1 \text{ square foot} = \left( \frac{1}{3.28} \right)^2 \text{ square metres} \approx 0.092903 \text{ square metres}$
Then, to convert 10 million square feet to square metres:

$10,000,000 \text{ square feet} \times 0.092903 \text{ square metres/square foot} \approx 930,000 \text{ square metres}$
Expressing this in scientific notation:

$930,000 = 9.3 \times 10^5 \text{ square metres}$

Thus, the answer is $9.3 \times 10^5$ .

Step 2

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.

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Answer

To find the total amount of paint required, divide the total area by the coverage per litre:

$\text{Total litres of paint required} = \frac{930,000 \text{ square metres}}{5 \text{ square metres/litre}} = 186,000 \text{ litres}$
Since the paint comes in 25 litre tins, we calculate the minimum number of tins required:

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

Thus, the minimum number of tins needed is 7440.

Step 3

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

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Answer

Rewrite $8$ as $2^3$ , so:

$8^{2x} = (2^3)^{2x} = 2^{6x}$
Thus, the equation becomes:

$2^{x-1} = 2^{6x}$
Equating the exponents:

$x - 1 = 6x$
Rearranging gives:

$-1 = 5x \implies x = -\frac{1}{5}$

Thus, the solution is $x = -\frac{1}{5}$ .

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

Given that 1 metre is equal to 3.28 feet, convert the surface area of the bridge into square metres

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.

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

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