Question Video: Solving a Real-World Problem Using Exponents Mathematics • Higher Education

On July 5, green algae was found on the bottom of a swimming pool whose width is 6 m and length is 12 m. If the area, in mm², the algae covers 𝑡 days later is given by 𝐴 = 4.3 ⋅ 2^(𝑡/3), when will the algae completely cover the bottom of the swimming pool?

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Video Transcript

On July 5th, green algae was found on the bottom of a swimming pool whose width is six meters and length is 12 meters. If the area, in millimeters squared, the algae covers 𝑡 days later is given by 𝐴 is equal to 4.3 times two to the power of 𝑡 over three, when will the algae completely cover the bottom of the swimming pool?

We’re given a real-world problem about green algae covering the bottom of a swimming pool. First, we’re given an equation to tell us the area of the swimming pool the green algae will cover 𝑡 days after July 5th. The area 𝐴 is equal to 4.3 times two to the power of 𝑡 over three. It’s important to note that this is in millimeters squared. We need to use this equation to determine when the algae will completely cover the bottom of the swimming pool.

To do this, we’re going to use the fact that we’re told the width of the swimming pool is six meters and the length of the swimming pool is 12 meters. So, let’s sketch a diagram of what’s happening. The width of our swimming pool is six meters, and the length of the swimming pool is 12 meters. And the algae covers an area of the bottom of the swimming pool given by 𝐴. Of course, we know 𝐴 is measured in millimeters squared, so it would make sense to also measure the length and width of the swimming pool in millimeters.

To do this, we need to recall that there are 1,000 millimeters in one meter. So, we need to multiply our length and our width by 1,000. This gives us the width of the swimming pool is 6,000 millimeters and the length of the swimming pool is 12,000 millimeters. We want to know how long it would take the algae to cover up all of the swimming pool. In other words, we need 𝐴 to be equal to the entire area of the swimming pool.

And we can just calculate the area of the swimming pool. It’s the length times the width, 6,000 millimeters multiplied by 12,000 millimeters. And if we calculate this, we get 72 million millimeters squared. So, the algae will have completely covered up the bottom of the pool if 𝐴 is equal to 72 million. So, we could just substitute 𝐴 is equal to 72 million into the equation given to us in the question. We could then solve this equation for 𝑡.

So, we’ll substitute 𝐴 is equal to 72 million into this equation. This gives us 72 million is equal to 4.3 times two to the power of 𝑡 over three. Remember, 𝑡 represents the number of days after July the 5th. So, we want to solve this equation for 𝑡. To start, we’ll divide both sides of our equations through by 4.3. This gives us 16,744,186.05 is equal to two to the power of 𝑡 over three. And we can see this is an exponential equation in 𝑡. So, to solve this, we’ll need to take logarithms of both sides of the equation.

We could solve this by taking a lot of different logarithms. However, it’s easiest to use logarithm base two. So, we’ll just take the logarithm base two of both sides of the equation. This gives us the log base two of 16,744,186.05 is equal to the log base two of two to the power of 𝑡 over three. And we know that two to the power of a number and taking the log base two of a number are inverses. So, the log base two of two to the power of 𝑡 over three is just equal to 𝑡 over three. And this means we can find an equation for 𝑡 by just multiplying through by three.

This gives us that 𝑡 is equal to three times the log base two of 16,744,186.05. And to the nearest number of days, this is equal to 72. Remember, 𝑡 represents the number of days after July the 5th. So, we need to work out which day is 72 days after July the 5th.

First, we know there are 31 days in July. This means 27 days after July the 5th will be August the 1st. One way of seeing this is 26 days after July the 5th is July the 31st. One more day after this will be August the 1st. Next, August also has 31 days. So, 31 days after August the first will be September the 1st. And we’ll add 27 and 31 together to give us 58. So, we’ve shown so far September the 1st is 58 days after July the 5th.

Remember, we need to go 72 days, so we need to go 14 more days after September the 1st. So, we just add 14 days onto September the 1st to get September the 15th. And this means 72 days after July the 5th is September the 15th. In this question, we were given a real-world problem about algae growing on the bottom of a swimming pool. We were able to use our formula to approximate on which date the algae would completely cover the bottom of the swimming pool. We estimated that the algae would completely cover the bottom of the swimming pool on September the 15th.

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