A country has 135 years of iron reserves at the current rate of consumption. Suppose that the demand for iron in the country is 10,000,000 tons per year. Furthermore the iron is consumed at a rate that increases 7.5% per year.
(a) How long will the iron reserves last?
(b) Suppose a recent discovery has doubled the quantity of iron reserves. Now how long will the iron reserves last?
To calculate how long the iron reserves will last, we need to consider the current rate of consumption and the rate at which the consumption is increasing.
(a) The current rate of consumption is 10,000,000 tons per year, and it increases by 7.5% per year.
To find out how long the iron reserves will last, we can use the formula for compound interest:
\[A = P(1 + r)^n\]
where:
A = the future value (in this case, the total consumption after n years)
P = the initial value (in this case, the current consumption rate)
r = the interest rate (in this case, the rate of increase in consumption)
n = the number of years
Let's calculate the number of years it will take for the reserves to be depleted:
135 = 10,000,000(1 + 0.075)^n
Dividing both sides of the equation by 10,000,000, we get:
0.0135 = (1.075)^n
Taking the natural logarithm of both sides, we have:
ln(0.0135) = n * ln(1.075)
Now we can solve for n:
n = ln(0.0135) / ln(1.075)
Using a calculator, we find that n ≈ 104.893.
Therefore, the iron reserves will last approximately 104.893 years.
(b) If the recent discovery has doubled the quantity of iron reserves, we now have 2 * 135 = 270 years' worth of reserves.
Using the same formula as before, but with the new consumption rate:
270 = 10,000,000(1 + 0.075)^n
Dividing both sides by 10,000,000, we get:
0.027 = (1.075)^n
Taking the natural logarithm of both sides:
ln(0.027) = n * ln(1.075)
Now we can solve for n:
n = ln(0.027) / ln(1.075)
Using a calculator, we find that n ≈ 190.501.
Therefore, with the doubled quantity of reserves, the iron reserves will last approximately 190.501 years.
To answer these questions, we'll use basic calculations and apply the concept of compound interest. Here's how you can calculate the answers:
(a) To determine how long the iron reserves will last, we need to find the rate at which the yearly consumption increases. According to the information given, the consumption rate increases by 7.5% per year. Therefore, the yearly consumption for each subsequent year can be calculated using the formula:
Yearly Consumption = Initial Consumption * (1 + Rate of Increase/100)
Let's calculate the consumption for the first few years to see the trend:
Year 1: 10,000,000 tons
Year 2: 10,000,000 tons * (1 + 7.5/100)
Year 3: Year 2 consumption * (1 + 7.5/100)
Year 4: Year 3 consumption * (1 + 7.5/100)
...
By using this pattern, we can calculate the consumption for subsequent years. Now, we need to determine when the consumption will reduce the iron reserves to zero.
Given that there are 135 years of iron reserves at the current consumption rate, we can set up an equation:
Total Reserves = Yearly Consumption * Number of Years
135 = 10,000,000 * (1 + 7.5/100)^Number of Years
Now, by solving this equation for "Number of Years," we can find out the time it will take to deplete the iron reserves.
(b) To answer the second question, we need to consider the recent discovery that has doubled the quantity of iron reserves. In this case, the new total reserves will be 270 years (135 years * 2). We can apply the same formula as before to determine the new time duration:
270 = Yearly Consumption * (1 + 7.5/100)^Number of Years
By solving this equation for "Number of Years," we can determine the new time it will take to deplete the iron reserves.
I hope this explanation helps you understand how to tackle this problem step by step.