WEBVTT
00:00:02.644 --> 00:00:09.784
A gold mine produced 2,257 kilograms in the first year but decreased 14 percent annually.
00:00:10.214 --> 00:00:15.174
Find the total amount of gold produced in the third year and the total across all three years.
00:00:15.204 --> 00:00:17.004
Give the answers to the nearest kilogram.
00:00:17.294 --> 00:00:20.224
In this question, we’re given some information about a gold mine.
00:00:20.294 --> 00:00:26.344
We’re told that in the first year of production, the gold mine produces 2,257 kilograms.
00:00:26.574 --> 00:00:31.044
But we’re told that year on year, this amount is decreasing by 14 percent.
00:00:31.174 --> 00:00:33.304
The question wants us to find two things.
00:00:33.334 --> 00:00:41.444
It wants us to find the amount of gold which is produced in the third year of production, and it wants us to find the total amount produced in all three of the first years.
00:00:41.654 --> 00:00:44.914
And we need to give both of our answers to the nearest kilogram.
00:00:45.164 --> 00:00:47.774
There’s actually two different ways we can answer this question.
00:00:47.804 --> 00:00:51.994
The first way is to directly find these values from the information given to us in the question.
00:00:52.184 --> 00:00:58.484
We’re told in the question, in the first year, the gold mine produces 2,257 kilograms of gold.
00:00:58.774 --> 00:01:06.194
We can find the amount of gold produced in the second year by remembering that this amount is going to decrease by 14 percent every year.
00:01:06.504 --> 00:01:10.014
There’s a few different ways of evaluating a decrease of 14 percent.
00:01:10.284 --> 00:01:13.744
One way is to multiply by one minus 0.14.
00:01:14.024 --> 00:01:18.464
And it’s worth pointing out here we’re subtracting 0.14 because this is a decrease.
00:01:18.464 --> 00:01:24.664
So we need to subtract, and we get 0.14 because our rate, 𝑟, is 14 and we need to divide this by 100.
00:01:25.034 --> 00:01:30.714
What we’re really saying here is a decrease in 14 percent is the same as multiplying by 0.86.
00:01:31.074 --> 00:01:39.204
Therefore, the amount of gold produced in the mine in the second year is 2,257 multiplied by 0.86 kilograms.
00:01:39.564 --> 00:01:44.954
And we can evaluate this exactly; we get 1941.02 kilograms.
00:01:45.104 --> 00:01:49.614
And we shouldn’t round our answer until the very end of the question, so we’ll leave this in exact form.
00:01:49.924 --> 00:01:52.454
We’re then going to want to do exactly the same for the third year.
00:01:52.654 --> 00:01:59.544
Once again, from the question, we know that the mine is going to produce 14 percent less gold in the third year than it did in the second year.
00:01:59.804 --> 00:02:05.574
So, one thing we could do is multiply the amount of gold we got in the second year by 0.86.
00:02:05.764 --> 00:02:09.964
However, it’s actually easy to just multiply our expression by 0.86.
00:02:10.204 --> 00:02:18.584
Multiplying this expression by 0.86 and simplifying, we get 2,257 multiplied by 0.86 squared kilograms.
00:02:18.894 --> 00:02:25.064
Calculating this expression exactly, we get 1669.2772 kilograms.
00:02:25.294 --> 00:02:28.004
We can now use these three values to answer our question.
00:02:28.134 --> 00:02:33.174
First, we can find the amount of gold produced in the third year by rounding this number to the nearest kilogram.
00:02:33.334 --> 00:02:36.614
This would then give us 1,669 kilograms.
00:02:36.844 --> 00:02:41.874
Next, we can find the total amount of gold produced in three years by adding these three values together.
00:02:42.364 --> 00:02:53.894
This gives us 2,257 kilograms plus 1941.02 kilograms plus 1669.2772 kilograms.
00:02:54.244 --> 00:03:00.514
And if we evaluate this expression, we get 5867.2972 kilograms.
00:03:00.834 --> 00:03:08.414
And to the nearest kilogram, we can see our first decimal place is two, so we need to round down, giving us 5,867 kilograms.
00:03:08.674 --> 00:03:12.584
However, what would have happened if we needed to find even more years of production?
00:03:12.734 --> 00:03:17.454
We can see that this method only really worked because we only had to calculate the first three years.
00:03:17.634 --> 00:03:22.024
If we were asked to find even more years in our example, we would need to notice something interesting.
00:03:22.134 --> 00:03:26.134
Each year we’re multiplying by a constant ratio of 0.86.
00:03:26.224 --> 00:03:32.734
And remember, in a sequence, if we’re multiplying by a constant ratio to get the next term in our sequence, we call this a geometric sequence.
00:03:32.764 --> 00:03:42.284
So, the gold produced in our mine after 𝑛 years forms a geometric sequence with initial value 𝑎, 2,257 kilograms, and ratio 𝑟, 0.86.
00:03:42.494 --> 00:03:51.144
We can then use what we know about geometric sequences to find the amount of gold produced after 𝑛 years in our mine and the total amount of gold produced after 𝑛 years.
00:03:51.324 --> 00:03:57.334
We just substitute 𝑛 is equal to three and our values for 𝑎 and 𝑟 into the two formula to find these expressions.
00:03:57.554 --> 00:04:00.034
And after rounding, we get the same answers we had before.
00:04:00.064 --> 00:04:07.704
𝑎 sub three will be 1,669 kilograms and 𝑆 sub three will be 5,867 kilograms.