QUESTIONScenario: The StageInstructions:View the video found on page 1 of this Journal activity.Using the information provided in the video, answer the questions below.Show your work for all calculations.Analyze The Students' ConjecturesMalcolm wants to build an outdoor stage with a total area of 350 square feet. The length of the stage should be 3 feet shorter than the width. He calculated the equation to be w2 − 3w = 3501. Complete the table to summarize each student's suggestion for figuring out the equation: (2 points: 1 point for each row of the chart)Student ConjectureMalcolm's bandmate Malcolm 2. Malcolm and his bandmate have different ideas for figuring out the equation. Which one do you think will make it easiest to solve the equation? Why? (2 points)3. Malcom's bandmate starts by making a factor table for w2 – 3w – 350 = 0.He is looking for the factors (w + p)(w + q) = 0, where p • q = –350 and p + q = –3. Fill in the last row of the table with different factors, p and q, so that p • q = –350. (2 points).p q p + q10 –35 –25–10 35 2550 –7 43–50 7 –43_____ _____ _____4. The bandmate’s factor table is not complete. It does not contain all the factors of –350.Does the factoring table contain the factors that can be used to solve w2 – 3w – 350 = 0? Explain your reasoning. (1 point)5. To make a perfect square trinomial, Malcolm said the rule for figuring out the number to add is . Is he correct? If not, what is the rule? (1 point)6. Calculate the number that you need to add to each side of the equation w2 – 3w = 350 to create a perfect square trinomial. Add this number to each side of the equation. Show your work. (3 points)7. Factor the trinomial. (3 points)8. Now that you've factored the equation, find the square root of each side and solve for w. Show your work and both solutions. (2 points)9. Do both of these solutions make sense in terms of the problem? Why or why not? (2 points)10. What are the maximum dimensions of the stage that Malcolm can build? (Round to the nearest foot.) (2 points)w = _____ feet l = _____ feetPROMPTAudio:O.K., everybody, it's "Do-It-Yourself" time! They say all it takes is the right tools, a little determination, and a good plan. Well, I've got the lumber, I've got the power saw — I've even got the elbow grease! The only thing I don't have is a plan. [The images change to show a young man carrying a plank of lumber, then a circle saw, then of an up close shot of a nail with a hammer about to hit it and a person in the background driving the hammer.]Here's the situation: My rock band got permission from the community center to build an outdoor wooden stage so we can perform some concerts this summer. [A park-like area with lots of trees is shown with benches and a small wooden stage.]I used all my powers of charm and persuasion to get a local lumberyard to donate some lumber. [A large red truck is shown filled with lumber.]We have enough materials to build a stage with a total area of 350 square feet. But because of the way our band does our setup, the length of the stage should be three feet shorter than the width. [A grass area is animated with a dashed-line outline of a square with "350 ft2." Wood is then shown inside the square but the right side elongates and "l = w – 3" is shown on the right side of the rectangle and "w" on the bottom of the rectangle.]Based on that info, I want to build the biggest platform I can. I think I can figure out the dimensions using a quadratic equation. (I know what you're thinking: He's musically talented and good at algebra? Wow!)Here's what I've come up with so far:We know that width times length equals total area. ["w • l = area" appears on-screen.] In this case, the length equals the width minus 3. So, width times width minus 3 should equal 350. [The following text appears on-screen: "l = w – 3; w(w – 3) = 350; w2 – 3w = 350"]That means: w squared minus 3w equals 350. I've been arguing with my bandmate about what to do next.[Spoken by a different voice:] Hey, it's not an argument, it's just a lively discussion!Student: So what's your suggestion?[Spoken by a different voice:] I think we need to subtract 350 from both sides, and then factor the equation. [w2 – 3w – 350 = 0 appears on-screen.]Student: I think it would be easier to factor if we change the equation so that it has a perfect square trinomial.[Spoken by a different voice:] That sounds too complicated.Student: We just have to figure out what number will complete the square on the left side of the equation, and then add that number to both sides of the equation. I think the rule for figuring out the number is b divided by 2, squared. ["" is shown on screen.]
Accepted Solution
A:
1. For this problem we just refer to the descriptions that you placed under the prompt. According to Malcolm's bandmate, it would be easier to solve the trinomial by subtracting 350 from both sides and then factoring the equation. Malcolm, on the other hand, thinks that we should manipulate the equation in order to make it a perfect square trinomial.
2. This trinomial would be easily solved by using Malcolm's idea. As Malcolm pointed out, you just need to apply a formula to manipulate the equation then you can find the roots in no time. Finding the factors of 350 just to solve the trinomial would be the hard way to go since you would be considering a lot of them.
3. For this item, we are just tasked to follow what Malcolm's bandmate started doing. So, we would just need to think of two numbers that would result to -350 when multiplied. To start off, let's think of something we can divide 350 by, let's say 70. Now, if we divide -350 by 70 the result would be -5 therefore that would be our two numbers (p and q). p + q would therefore just be 65.
p = 70 q = -5 p + q = 65
4. No, the factor table is not complete since you would need factors of -350 that would add up to -3, the coefficient of w. This rule is sort of a shortcut when factoring the trinomial, since expressing the roots in the form of (w + p)(w + q) would lead you to the original expression by following the rules. We do not see any factors that add up to -3 in the factor table.
5. Malcolm was almost right, but technically he missed the initial step. To make a perfect square trinomial, you first need to make sure that a (or the coefficient of the leading term) is equal to one. If not, you first need to divide the entire equation by a. Then, you apply what Malcolm says.
6. Since the coefficient of the leading term is already equal to one, we do not need to worry about the initial step anymore. For this item, we just need to divide -3 by two and square it then add the resulting number to both sides. The solution is shown below:
7. To factor the trinomial, we just really need to make sense of the process we just did in the previous item. Notice that we just squared -1.5, after dividing -3 by 2. If you look at it closely, this is just the process of expanding the square of a binomial in reverse. Therefore, we know that our resulting expression is the square of (w - 1.5)
[tex](w-1.5)^{2}=352.25 [/tex]
8. For this item we are just simply tasked to follow the instructions given. The process described in the item is how we are supposed to solve for the variable. The work and solution for this item is shown below:
9. The solution does make sense in terms of the problem because, firstly, the answer we have arrived on is a positive number which means it is a realistic value of a measurement, and secondly, our solution in the previous item just followed basic arithmetic so we did not violate anything in the problem.
10. For this item, we can find the length by subtracting the value of the width by 3, as was dictated in the prompt. But before this, we first need to round our answer for the width to the nearest foot. 20 minus 3 is just 17. Multiplying 20 and 17 might not get us to 350 but this is just an approximation anyway. (Getting the product of the numbers before rounding off would give us an accurate one.)