A fish pond initially has a population of 300 fish. When there is enough fish food, the population,P, of fish grows as a function of time, t, in years, as P(t)= 300(1.05)t. The initial amount of fish food in the pond is 1000 units, where 1 unit can sustain one fi sh for a year. The amount, F, of fish food is decreasing according to the function F(t) =1000(0.92)t
a) Graph the functions P(t) and F(t) on the same set of axes. Describe the
nature of these functions.
b) Determine the mathematical domain and range of these functions.
c) Identify the point of intersection of these two curves. Determine the coordinates, to two decimal places, and explain what they mean. Call this point in time the crisis point.
d) Graph the function y =F(t)-P(t). Explain the significance of this function.
e) What is the t-intercept, to two decimal places, of the function y = F(t) - P(t)? How does this relate to the crisis point?
f) Comment on the validity of the mathematical model for P(t) for t-values greater than this intercept. Sketch how you think the curve should change in this region. Justify your answers.
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Just give yourself a nick-name so we can tell one post from another
www.jiskha.com/questions/1882777/consider-a-pond-that-contains-an-initial-population-of-300-fish-when-there-are-enough
no haha, so many grammar mistakes in that one. It didn't even make sense ..? @mathhelper
a) To graph the functions P(t) and F(t) on the same set of axes, we can plot points using different values of t.
For P(t):
- Choose different values for t, such as 0, 1, 2, 3, etc.
- Substitute each value into the equation P(t) = 300(1.05)^t to calculate the corresponding population.
- Plot these points on the graph.
For F(t):
- Follow the same process, but using the equation F(t) = 1000(0.92)^t to calculate the corresponding amount of fish food.
- Plot these points on the same graph.
The nature of these functions can be described as follows:
- P(t) represents the exponential growth of the fish population over time. As t increases, P(t) increases at an accelerating rate.
- F(t) represents the exponential decay of the fish food over time. As t increases, F(t) decreases at a decelerating rate.
b) The mathematical domain refers to the set of valid input values (t-values) for a function, while the range is the set of corresponding output values.
For P(t):
- The domain is all real numbers since time (t) can be any positive or negative number.
- The range is all positive real numbers since the population (P(t)) cannot be negative.
For F(t):
- The domain is all real numbers since time (t) can be any positive or negative number.
- The range is all positive real numbers since the amount of fish food (F(t)) cannot be negative.
c) To determine the point of intersection, we need to find the values of t where P(t) = F(t).
- Set P(t) equal to F(t): 300(1.05)^t = 1000(0.92)^t.
- To solve this equation, we can use numerical methods such as graphing calculators or spreadsheet software.
- The coordinates of the point of intersection represent the time (t) and the corresponding population (P(t)) or amount of fish food (F(t)) at that point.
- This point is called the crisis point because it is where the fish population and the availability of fish food intersect, indicating a critical balance.
d) To graph the function y = F(t) - P(t), we need to subtract the value of P(t) from F(t) for each value of t.
- Calculate the difference F(t) - P(t) for different values of t.
- Plot these points on a graph.
- The significance of this function is that it represents the net difference between the amount of fish food and the fish population at any given time. If the value is positive, it means there is excess fish food, whereas a negative value indicates a shortage of fish food.
e) To find the t-intercept of the function y = F(t) - P(t), we need to determine the value of t at which y (or F(t) - P(t)) is equal to zero.
- Set F(t) - P(t) equal to zero: 1000(0.92)^t - 300(1.05)^t = 0.
- Solve this equation to find the value of t where the function crosses the t-axis.
- The t-intercept represents the point in time when the amount of fish food is equal to the fish population, indicating a critical balance or crisis point.
f) For t-values greater than the t-intercept, the validity of the mathematical model for P(t) may be questionable.
- As t increases beyond the t-intercept, the equation P(t) = 300(1.05)^t would suggest that the fish population continues to grow exponentially.
- However, this may not be realistic as the availability of fish food is limited, and the fish population cannot grow indefinitely without sufficient resources.
- Therefore, in this region, it is likely that the curve for P(t) should start to level off or decline rather than continuing to increase exponentially.
- The exact shape of the curve beyond the t-intercept would depend on various factors and would require further analysis or adjustments to the model. A possible sketch could show a flattening or a decline in the curve.