Video Transcript
The value of an antique vase increases at a rate of seven percent a year. If the vase was valued at 1,200 dollars five years ago, what would its current value be? Give your answer to the nearest ten dollars if necessary.
So in order to solve a problem like this, what we want to do is set up an equation. And it’s gonna be an exponential equation. And we’re gonna look at a general form here. And that is that the function of 𝑥 is equal to 𝐴, which is our initial value, multiplied by 𝑏, where 𝑏 is something that tells us about the rate — and usually it’s in the form of the decimal multiplier that the rate gives us, always remembering as well that 𝑏 must be positive and not equal to one — and then this is raised to 𝑥, which is our independent variable. And this is usually time, so the number of time periods that we’re looking at.
So to solve a problem like this, what we usually do is find out if we have each of the parts that we need to set up our equation. So first of all, we want our initial value 𝐴, and this is going to be equal to 1,200 dollars. And then our 𝑏 is gonna be equal to 1.07. And the reason that is is if we imagine that we had 100 percent, this means 100 out of 100. So it’d give us one. And then what we wanted to do is increase this by seven percent. So when we add seven percent on, what we get is 107 percent. What this means is 107 out of 100. So this is gonna give us 1.07, which is our decimal multiplier. So if we in fact want to increase anything by a rate of seven percent, then what we can do is multiply it by 1.07.
And then, finally, we have 𝑥 is equal to five. And it’s five because we know that the vase was valued at 1,200 dollars five years ago, and we want to find out the value now. And also we’d know that we don’t have to do anything different with the time periods because the rate of increase is also per year. So therefore, what we can say is that the value of the vase is gonna be equal to 1,200 multiplied by 1.07 to the power of five. So therefore, we can say that the value of the vase is going to be 1,683.02 [1,683.062] et cetera dollars.
However, we haven’t quite finished the question there. And that’s because if we look back at the question, we’re asked to round our answer to the nearest 10 dollars where necessary. So therefore, what we can say is that the current value of the vase to the nearest 10 dollars is 1,680 dollars.
