Given a sheet of paper 8.5 by 11, fold the top left comer down to a...

   

Transcribed Image Text:

Given a sheet of paper 8.5" by 11, fold the top left comer down to a point on the bottom edge. Where should you place this corner to maximize the area of the triangle formed in the bottom left corner? Remember, the area of a triangle can be given by the geometry formula: Area = This formula tells us that the area is a function of the and the of the triangle. Let's do some measuring (and calculating)) and determine the area for some triangles. Take a blank sheet of 8.5" by 11" piece of paper and (holding it sideways) mark the bottom edge and the left side of the paper in 0.5* units using a ruler. You only need to mark the bottom edge up to 8.5". Fold the top left corner down to the point on the bottom edge that measures 1". Estimate the height of the triangle to the nearest 0.1". Now calculate the area of the triangle. We are now going to let the base of the triangle grow longer. What will happen to the height of the triangle? What do you think will happen to the area of the triangle? Let's see if the above conjectures are correct. Fold the top left corner down to the indicated point (in the table below) on the bottom edge of the piece of paper. Estimate the height of each triangle (to the nearest 0.1 in). Then calculate the area of the triangle. Use the table below to show your results. Base Height Area Base Height Area 5" 2" 6″ 3" 7" 8° From your chart where does it appear that the maximum value of the area of the triangle occurs? The maximum area of the triangle is and it occurs when the base is To determine the exact value of the base that will maximize the area we need to write a function for the area in terms of the base. In order to do this we first need to write a function for the height of the triangle in terms of the base. (This is not easy!). Let's look at this data in graph form. Put the values of the base in L1 and the values of the height in L2. Turn on Plot 1 from your Stat Plot menu and select the scatterplot option. Set up your window to get a good view of your data points and plot the data. Draw a diagram of the data points below and indicate the window you used on your calculator. Dxonin Xmax): Exmoin. Ymax): What type of function do you think your data matches the closest? Using the regression option see if your guess was correct? Graph your regression equation. How does it fit? Let's try to find this relationship algebraically. (Remember, this is the relationship between the height of the triangle and the base of the triangle.) Draw a diagram of the paper with the top left corner folded down (similar to the figure on the other side). Label the base of the triangle B. Label the height of the triangle H. What is the length of the hypotenuse of the triangle? Your answer should be in terms of H, not B. (Hint: Unfold your paper!) Hypotenuse Use the Pythagorean Theorem and write an equation which shows the relationship among these three sides. .)² Solve this equation for H. H = Put the equation that you found for H into Y1 and graph it. You will be letting "x" represent the variable "B" and "y" represent the variable "H". Does it fit your data points? It should if you found it correctly!) We are now ready to write our equation for the area of the triangle. The area of the triangle (in terms of the base B) is A = The independent variable is and the dependent variable is Before we graph this function let's change our data points in L2 to show the relationship between the base of the triangle and the area of the triangle. You have these values in your AREA column on the other side. Graph them and show your graph below. Now put your area equation in Y1 of your calculator and see how it fits your data. Hopefully it fits very well! Use your calculator to find the value of the base that maximizes the area of the triangle. | The maximum occurs when the base is and the height is T Given a sheet of paper 8.5" by 11, fold the top left comer down to a point on the bottom edge. Where should you place this corner to maximize the area of the triangle formed in the bottom left corner? Remember, the area of a triangle can be given by the geometry formula: Area = This formula tells us that the area is a function of the and the of the triangle. Let's do some measuring (and calculating)) and determine the area for some triangles. Take a blank sheet of 8.5" by 11" piece of paper and (holding it sideways) mark the bottom edge and the left side of the paper in 0.5* units using a ruler. You only need to mark the bottom edge up to 8.5". Fold the top left corner down to the point on the bottom edge that measures 1". Estimate the height of the triangle to the nearest 0.1". Now calculate the area of the triangle. We are now going to let the base of the triangle grow longer. What will happen to the height of the triangle? What do you think will happen to the area of the triangle? Let's see if the above conjectures are correct. Fold the top left corner down to the indicated point (in the table below) on the bottom edge of the piece of paper. Estimate the height of each triangle (to the nearest 0.1 in). Then calculate the area of the triangle. Use the table below to show your results. Base Height Area Base Height Area 5" 2" 6″ 3" 7" 8° From your chart where does it appear that the maximum value of the area of the triangle occurs? The maximum area of the triangle is and it occurs when the base is To determine the exact value of the base that will maximize the area we need to write a function for the area in terms of the base. In order to do this we first need to write a function for the height of the triangle in terms of the base. (This is not easy!). Let's look at this data in graph form. Put the values of the base in L1 and the values of the height in L2. Turn on Plot 1 from your Stat Plot menu and select the scatterplot option. Set up your window to get a good view of your data points and plot the data. Draw a diagram of the data points below and indicate the window you used on your calculator. Dxonin Xmax): Exmoin. Ymax): What type of function do you think your data matches the closest? Using the regression option see if your guess was correct? Graph your regression equation. How does it fit? Let's try to find this relationship algebraically. (Remember, this is the relationship between the height of the triangle and the base of the triangle.) Draw a diagram of the paper with the top left corner folded down (similar to the figure on the other side). Label the base of the triangle B. Label the height of the triangle H. What is the length of the hypotenuse of the triangle? Your answer should be in terms of H, not B. (Hint: Unfold your paper!) Hypotenuse Use the Pythagorean Theorem and write an equation which shows the relationship among these three sides. .)² Solve this equation for H. H = Put the equation that you found for H into Y1 and graph it. You will be letting "x" represent the variable "B" and "y" represent the variable "H". Does it fit your data points? It should if you found it correctly!) We are now ready to write our equation for the area of the triangle. The area of the triangle (in terms of the base B) is A = The independent variable is and the dependent variable is Before we graph this function let's change our data points in L2 to show the relationship between the base of the triangle and the area of the triangle. You have these values in your AREA column on the other side. Graph them and show your graph below. Now put your area equation in Y1 of your calculator and see how it fits your data. Hopefully it fits very well! Use your calculator to find the value of the base that maximizes the area of the triangle. | The maximum occurs when the base is and the height is T

Answer rating: 100% (QA)

Sol Given that Area xBxH ie area is fanction of and hight of the base the 4 from t... View the full answer