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Chloe bought an antique vase for 600 dollars.
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The value of the vase increases by four percent each year.
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Write an equation that can be used to find the value of the vase in dollars, 𝐴, 𝑡 years after it was purchased.
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Well, to solve this problem, what we’re going to do is set up an exponential equation.
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And we, in fact, have a general form that can help us to do that.
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And that is that 𝑓 of 𝑥 is equal to 𝐴 multiplied by 𝑏 to the power of 𝑥, and that’s where 𝐴 is the initial amount, 𝑏 is our rate, which we have to write as the decimal multiplier, then we’ve got our 𝑥, which is the independent variable which is usually our number of time periods.
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Okay, great.
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But how does this apply to our problem?
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Well, in our problem, our 𝑓 of 𝑥 is going to be 𝐴 because this is what we’re trying to find cause we want to find the value of the vase in dollars, which is our 𝐴.
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So, we’re going to have 𝐴 equals.
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And then the 𝐴 from the general formula, so the initial amount, is also told to us in the question because we can see that the initial amount is the amount that Chloe bought the antique vase for, and that is 600 dollars.
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Now, the next part, which is the 𝑏 from the general form or our rate, we’re gonna have to work out cause what we’re told is that the value of the vase increases by four percent each year.
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So, let’s try and work out what an increase of four percent would be as a decimal multiplier.
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Well, let’s think if we begin with 100 percent of something and we increase it by four percent, then what we’re gonna do is add on four percent.
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So, this will give us a new percentage of 104 percent.
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Well, 104 percent means 104 out of 100.
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So, what that’s gonna tell us is that it’s 104 divided by 100, which as a decimal is gonna be 1.04.
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So, therefore, we can say that the decimal multiplier for increasing by four percent is gonna be 1.04.
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So, that’s gonna be our rate or our 𝑏 from the general form.
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Well, finally, what we want to find is our 𝑥, which is our independent variable, which we said was gonna be the time period.
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Well, we’re told that the time period is years, and we’re representing it with 𝑡.
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So, therefore, our 𝑥 from the general form is going to be 𝑡 in our equation.
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So, therefore, we can say that if Chloe bought an antique vase for 600 dollars and the value of that vase increased by four percent each year, then the equation that we could use to find the value of the vase in dollars, 𝐴, 𝑡 years after it was purchased would be 𝐴 is equal to 600 multiplied by 1.04 to the power of 𝑡.