There are oniginally 245 foxes and 124 rabbits on a particular game reserve. The
fox pooulation at a rate of 27 foxes per year, and the rabbits increase at a
rate of 52 rabbits per year. Under these conditions, how long does it take for the
number of rabbits to catch up with the number of foxes? (Round your answer to
kwo decimal places.)
yr
How many of each animal will be present at that time? (Round your answers
down to the nearest whole animal.)
fowes
rabbits

Question

There are oniginally 245 foxes and 124 rabbits on a particular game reserve. The
fox pooulation

at a rate of 27 foxes per year, and the rabbits increase at a
rate of 52 rabbits per year. Under these conditions, how long does it take for the
number of rabbits to catch up with the number of foxes? (Round your answer to
kwo decimal places.)
yr
How many of each animal will be present at that time? (Round your answers
down to the nearest whole animal.)
fowes
rabbits

UpStudy AI · Accepted Answer

**Step 1. Set Up the Equations**

The number of foxes after $$ t $$ years is given by:

$$
\text{Foxes}(t) = 245 + 27t
$$

The number of rabbits after $$ t $$ years is given by:

$$
\text{Rabbits}(t) = 124 + 52t
$$

**Step 2. Equate the Two Expressions**

To find when the number of rabbits catches up with the number of foxes, set the two expressions equal:

$$
245 + 27t = 124 + 52t
$$

**Step 3. Solve for $$ t $$**

Subtract $$ 27t $$ from both sides:

$$
245 = 124 + 25t
$$

Subtract $$ 124 $$ from both sides:

$$
121 = 25t
$$

Divide both sides by $$ 25 $$:

$$
t = \frac{121}{25} \approx 4.84 \text{ years}
$$

**Step 4. Determine the Number of Animals at Time $$ t $$**

Substitute $$ t \approx 4.84 $$ back into one of the equations (using the foxes’ equation):

$$
\text{Foxes}(t) = 245 + 27(4.84) \approx 245 + 130.68 \approx 375.68
$$

Since we round down to the nearest whole animal, there will be:

$$
375 \text{ foxes}
$$

For the rabbits:

$$
\text{Rabbits}(t) = 124 + 52(4.84) \approx 124 + 251.68 \approx 375.68
$$

Rounded down, there will be:

$$
375 \text{ rabbits}
$$

**Final Answers:**

- It takes approximately $$ 4.84 $$ years for the rabbits to catch up with the foxes.
- At that time, there will be $$ 375 $$ foxes and $$ 375 $$ rabbits.
It takes approximately 4.84 years for the number of rabbits to catch up with the number of foxes. At that time, there will be 375 foxes and 375 rabbits.

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