Given:
The table of values.
To find:
The function for the given table.
Solution:
From the given table it is clear that the y-values are same as the corresponding x-values. It means y is directly proportional to x with 1 as a constant of proportionality.
Mathematically, it can be defined as:
Substitute in the above equation to find the required function.
Therefore, the correct option is A.
The graph of the given function is attached showing characteristics of the function.
The correct responses are;
Reasons:
The given function is; f(t) = 10·(t³ - 12·t² + 45·t + 100)
Where;
t = Number of days
0 ≤ t ≤ 8
(a) The derivative of the given function is presented as follows;
F'(t) = 10 × (3·t² - 2×12·t + 45) = 10·(3·t² - 24·t + 45)
Therefore;
F'(4) = 10 × (3 × 4² - 24 × 4 + 45) = -30
Therefore F'(4) = -30 means that on day 4, the <u>number of fish in the bay is decreasing at 30 fish per day</u>.
(b) The absolute minimum is given as follows;
At a minimum or maximum value, F'(t) = 10·(3·t² - 24·t + 45) = 0
Which gives;
3·(t - 5)·(t - 3) = 0
t = 5, or t = 3
At t = 5, we have;
f(5) = 10 × (5³ - 12 × 5² + 45 × 5 + 100) = 1,500
At t = 3, we have;
f(3) = 10 × (3³ - 12 × 3² + 45 × 3 + 100) = 1,540
Therefore, f(5) = 1,500 is a local maximum
However, at x = 0, we have;
f(0) = 10 × (0³ - 12 × 0² + 45 × 0 + 100) = 1000
At x = 8, we have;
f(8) = 10 × (8³ - 12 × 8² + 45 × 8 + 100) = 2,040
Therefore, the absolute minimum is given at t = 8, where f(t) = <u>1,000</u>
(c) The values of <em>t</em> at which the rate of change in the number of fish in the
bay is decreasing is between the local maximum at t = 3, and the local
minimum at t = 5, which gives;
The rate of change of the number of fish is decreasing for values of t in the range;
(d) P = 10·(t³ - 12·t² + 45·t + 100)
P' = 10·(3·t² - 24·t + 45)
At time t = c, we have;
<em>Based on a similar question online, we have;</em>
<em>(c) The interval over which the rate of change is decreasing</em>
<em>(d) The rate of change of the number of pelicans flying near the bay at t = c</em>
Learn more about differentiation of functions here:
