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denis-greek [22]
3 years ago
12

After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. You notice the rainbow is the shape of a parab

ola.
The equation for this parabola is y = -x2 + 36.

Courtesy of Texas Instruments


In the distance, an airplane is taking off. As it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. Create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow.
Analyze the two functions. Answer the following reflection questions in complete sentences.
What is the domain and range of the rainbow? Explain what the domain and range represent. Do all of the values make sense in this situation? Why or why not?
What are the x- and y-intercepts of the rainbow? Explain what each intercept represents.
Is the linear function you created with your table positive or negative? Explain.
What are the solutions or solution to the system of equations created? Explain what it or they represent.
Create your own piecewise function with at least two functions. Explain, using complete sentences, the steps for graphing the function. Graph the function by hand or using a graphing software of your choice (remember to submit the graph)

Mathematics
1 answer:
Gekata [30.6K]3 years ago
7 0

We can answer the first part of the question not taking intersecting function into account. The domain of y=-x^{2}+36 is all the numbers, x∈(-∞, +∞) and the range is y∈(-∞, 36]. We can observe these results with the help of a graph, as well. Since we are talking about the rainbow, the values above the ground level will make sense. In this case, we will take into account the range as it changes between 0 and 36, included and the domain between -6 and 6. Here (0;36) is the y-intercept and (-6;0) and (6;0) are the x-intercepts of the parabola.


Since in our problem, the linear function that intersects parabola is not given, we have to provide it by ourselves according to the conditions of the problem. It could be any line intersecting parabola in two points. One important point is that the y-intercept has to be no more than 36. Considering these conditions, we can set our linear function to be y=0.5x+5. We can observe the points that we included in the table (they have been given with orange dots in the graph and the table is attached below). We can see that the values of the function (values of y) are positive. Indeed, we are discussing the part of the rainbow above the ground level.


The system of equations with linear and quadratic functions has got two solutions and we can observe that result from the graph. The solutions are (-5.823; 2.088) and (5.323; 7.662). The solutions are the intersection points.

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Answer:

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Step-by-step explanation:

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Leto [7]

Answer:

The Depth of water in the bathtub initially = 2 cm

Step-by-step explanation:

For this question, we want to f8nd the amount of water present in the bathtub at time t = 0 using line of best fit, that is, linear regression analysis.

The table provided shows that the depth of water in the bathtub (y) changes as time (x) progresses.

Time (min) (x) | 2 | 4 | 6 | 8 | 10 | 12 | | 14

Depth (cm) (y) | 8 | 15 | 29 | 37 | 39 | 49 | 55

Running the analysis on a spreadsheet application, like excel, the table of parameters is obtained and presented in the first attached image to this solution.

Σxᵢ = sum of all the independent variables (sum of all the time data)

Σyᵢ = sum of all the dependent variables (sum of all the depth data)

Σxᵢyᵢ = sum of the product of each dependent variable and its corresponding independent variable

Σxᵢ² = sum of the square of each independent variable (time data)

Σyᵢ² = sum of the square of each dependent variable (depth data)

n = number of variables = 7

The scatter plot and the line of best fit is presented in the second attached image to this solution

Then the regression analysis is then done

Slope; m = [n×Σxᵢyᵢ - (Σxᵢ)×(Σyᵢ)] / [nΣxᵢ² - (∑xi)²]

Intercept b = [Σyᵢ - m×(Σxᵢ)] / n

Mean of x = (Σxᵢ)/n

Mean of y = (Σyᵢ) / n

Sample correlation coefficient r: r =

[n*Σxᵢyᵢ - (Σxᵢ)(Σyᵢ)] ÷ {√([n*Σxᵢ² - (Σxᵢ)²][n*Σyᵢ² - (Σyᵢ)²])}

And -1 ≤ r ≤ +1

All of these formulas are properly presented in the third attached image to this answer

The table of results; mean of x, mean of y, intercept, slope, regression equation and sample coefficient is presented in the fourth attached image to this answer.

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y = 3.911x + 1.857

y = 3.911(0) + 1.857 = 1.857 cm = 2 cm to the nearest integer.

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