Shannon manages a small zoo and she has been analyzing the attendance data. Shannon finds that the number of visitors increases
exponentially as the temperature increases, and this situation is represented by the function f(x) = 3x. Shannon also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by f(x) = −x + 4. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.
On the graph two functions are joined at (1,3) but I would like to know how to solve it with the visual graph.
On the graph two functions are joined at (1,3) but I would like to know how to solve it with the visual graph.
2 answers:
To plot the two functions, you need to assign values for x. By selecting various x values, corresponding y values are calculated. You then plot the points as shown. The blue line illustrates the function f(x) = 3x, while the orange line indicates f(x) = -x + 4.
These two lines intersect at the coordinates (1,3). To solve for this analytically, we establish the system of equations:
y = 3x
y = -x + 4
By setting these two equations for y equal to each other:
3x = -x + 4
4x = 4
x = 1
Next, substituting x into one of the original equations gives us: y = 3x = 3(1) = 3. Therefore, the solution is (1,3), meaning at a temperature of 3 units, the number of visitors entering matches those exiting the zoo, indicating a balance of one person entering for each one leaving.
These two lines intersect at the coordinates (1,3). To solve for this analytically, we establish the system of equations:
y = 3x
y = -x + 4
By setting these two equations for y equal to each other:
3x = -x + 4
4x = 4
x = 1
Next, substituting x into one of the original equations gives us: y = 3x = 3(1) = 3. Therefore, the solution is (1,3), meaning at a temperature of 3 units, the number of visitors entering matches those exiting the zoo, indicating a balance of one person entering for each one leaving.
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Response:
24.2 (rounded to the nearest tenth)
Detailed breakdown:
The data type involved is ungrouped
standard deviation =√ [∑ (x-μ)² / n]
average (μ)=∑x/n
=
=33.3
For the values 5, 5, 6, 12, 13, 26, 37, 49, 51, 56, 56, 84, the x-μ results will be
-28.3, -28.3, -27.3, -21.3, -7.3, 3.7, 15.7,17.7, 22.7,22.7,50.7
Calculating (x-μ)² gives: 800.89, 800.89, 745.29, 453.69, 53.29, 13.69, 246.49, 313.29, 515.29, 515.29, 2570.49
Then, ∑ (x-μ)² results in = 7028.59
Thus, standard deviation = √(7028.59 / 12)
=24.2
Answer:
Error made by Andrew: He identified incorrect factors based on the roots.
Step-by-step explanation:
The roots of the polynomial consist of: 3, 2 + 2i, 2 - 2i. By the factor theorem, if a is a root of the polynomial P(x), then (x - a) must be a factor of P(x). According to this premise:
(x - 3), (x - (2 + 2i)), (x - (2 - 2i)) represent the factors of the polynomial.
<pBy simplification, we obtain:(x - 3), (x - 2 - 2i), (x - 2 + 2i) as the respective factors.
This is where Andrew's mistake occurred. Factors should always be in the form (x - a), not (x + a). Andrew expressed the complex factors incorrectly, resulting in an erroneous conclusion.
Thus, the polynomial can be expressed as: