MULTIPLE REPRESENTATIONS In this problem, you will explore how dilations affect surface area. Use the cube...

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MULTIPLE REPRESENTATIONS In this problem, you will explore how dilations affect surface area. Use the cube at the right. a. Numerical Find the surface area of the cube. b. Tabular Copy and complete the table shown. Scale Factor of Dilation Length of Side Surface Area Original x2 x3 " " " c. Analytical Compare the original surface area to the surface area after a dilation by a scale factor of 2, then by a scale factor of 3. d. Verbal Does this same relationship exist with any rectangular prism? Explain your reasoning. 16. SHIPPING A shipping box in the shape of a rectangular prism can hold 576 cubic inches of material. The length of the box is 12 inches and the width of the box is 8 inches. a. Find the surface area of the box. b. Another box with the same volume has dimensions of 16 inches by 9 inches by 4 inches. Which box has the greater surface area? Explain. c. Predict the shape of a rectangular prism with the same volume but with a greater surface area than either of the boxes above. Then, test your prediction. 17. FINANCIAL LITERACY The boxes shown below are puzzle boxes that are made from wood. If the wood to make the boxes costs $1.30 per square inch, which box would cost more to make? Explain. Box 1 Box 2 Standa 23. A re volu mea 20 A Wh req A B C D 24. SH re ra an Spir Find th 27. ar 2.5 in 2.4 in. 1.4 in 38 in 2.4 in. 2.4 in. 30. Al P ta H.O.T. Problems Use Higher-Order Thinking Skills 18. OPEN ENDED Name the dimensions of a cube with a surface area between 10 and 20 square inches. 19. CHALLENGE A box manufacturer wants to make a box that has a volume of 216 cm and uses the least amount of cardboard possible. Find the dimensions of that box. 20. ERROR ANALYSIS Serena and Rhianna want to find the surface area of the cube shown. Is either of them correct? Explain. Eval 31. 33. Serena 2.2.2=8cm Rhianna 8.2+2-2-2-24 cm 2 cm Fin 35 21. CHALLENGE Sketch two prisms such that one has a greater volume and the other has a greater surface area. 22. EWRITING IN MATH Explain the difference between surface area and volume. 736 Lesson 12-5 | Surface Area of Prisms Problem 5 - Uncertainty and insurance (20 points) A farmer is facing an uncertain harvest outcome. If the weather is good, the value of the crop is AG = 100 and if the weather is bad, the value is AB = 25. With 80% probability the weather will be good, and with 20% probability it will be bad. Denote the probability of good weather by 0.8. The farmer's utility of the bundle A = (AG, AB) takes an expected utility form U (A)=AG + (1 - ) AB. 1. Calculate the certainty equivalent associated with bundle A. 2. Calculate the risk premium associated with bundle A. 3. Is the farmer risk loving, risk averse, or risk neutral? 2 4. An insurance company is offering to sell the farmer an insurance policy that provides perfect insurance. Specifically, in case of good weather it pays nothing and in case of bad weather it pays 75. The insurance company will sell this policy at premium M. (a) If the insurance company sets an actuarially fair price, what is M? (b) What is the farmer's willingness to pay for the insurance policy? 1. Suppose that the individual's utility function is given by u(x) = x. Find the expected value, expected utility, certainty equivalent, and risk premium for the following lottery: P: ($10,000, 0.4; $400, 0.6) 2. Assume that you are indifferent between getting $700 and getting $1,000 with probability 0.8 (and O with prob. 0.2). Also assume that you are indifferent between getting $300 and getting $700 with probability 0.6 (and 0 with prob. 0.4). Consider lottery A, which gives you $1,000 with probability 2/3 (and 0 with prob. 1/3), and lottery B, which gives you a 50-50 chance of $300 and $700. Which lottery do you prefer if you are an expected utility maximizer? MULTIPLE REPRESENTATIONS In this problem, you will explore how dilations affect surface area. Use the cube at the right. a. Numerical Find the surface area of the cube. b. Tabular Copy and complete the table shown. Scale Factor of Dilation Length of Side Surface Area Original x2 x3 " " " c. Analytical Compare the original surface area to the surface area after a dilation by a scale factor of 2, then by a scale factor of 3. d. Verbal Does this same relationship exist with any rectangular prism? Explain your reasoning. 16. SHIPPING A shipping box in the shape of a rectangular prism can hold 576 cubic inches of material. The length of the box is 12 inches and the width of the box is 8 inches. a. Find the surface area of the box. b. Another box with the same volume has dimensions of 16 inches by 9 inches by 4 inches. Which box has the greater surface area? Explain. c. Predict the shape of a rectangular prism with the same volume but with a greater surface area than either of the boxes above. Then, test your prediction. 17. FINANCIAL LITERACY The boxes shown below are puzzle boxes that are made from wood. If the wood to make the boxes costs $1.30 per square inch, which box would cost more to make? Explain. Box 1 Box 2 Standa 23. A re volu mea 20 A Wh req A B C D 24. SH re ra an Spir Find th 27. ar 2.5 in 2.4 in. 1.4 in 38 in 2.4 in. 2.4 in. 30. Al P ta H.O.T. Problems Use Higher-Order Thinking Skills 18. OPEN ENDED Name the dimensions of a cube with a surface area between 10 and 20 square inches. 19. CHALLENGE A box manufacturer wants to make a box that has a volume of 216 cm and uses the least amount of cardboard possible. Find the dimensions of that box. 20. ERROR ANALYSIS Serena and Rhianna want to find the surface area of the cube shown. Is either of them correct? Explain. Eval 31. 33. Serena 2.2.2=8cm Rhianna 8.2+2-2-2-24 cm 2 cm Fin 35 21. CHALLENGE Sketch two prisms such that one has a greater volume and the other has a greater surface area. 22. EWRITING IN MATH Explain the difference between surface area and volume. 736 Lesson 12-5 | Surface Area of Prisms Problem 5 - Uncertainty and insurance (20 points) A farmer is facing an uncertain harvest outcome. If the weather is good, the value of the crop is AG = 100 and if the weather is bad, the value is AB = 25. With 80% probability the weather will be good, and with 20% probability it will be bad. Denote the probability of good weather by 0.8. The farmer's utility of the bundle A = (AG, AB) takes an expected utility form U (A)=AG + (1 - ) AB. 1. Calculate the certainty equivalent associated with bundle A. 2. Calculate the risk premium associated with bundle A. 3. Is the farmer risk loving, risk averse, or risk neutral? 2 4. An insurance company is offering to sell the farmer an insurance policy that provides perfect insurance. Specifically, in case of good weather it pays nothing and in case of bad weather it pays 75. The insurance company will sell this policy at premium M. (a) If the insurance company sets an actuarially fair price, what is M? (b) What is the farmer's willingness to pay for the insurance policy? 1. Suppose that the individual's utility function is given by u(x) = x. Find the expected value, expected utility, certainty equivalent, and risk premium for the following lottery: P: ($10,000, 0.4; $400, 0.6) 2. Assume that you are indifferent between getting $700 and getting $1,000 with probability 0.8 (and O with prob. 0.2). Also assume that you are indifferent between getting $300 and getting $700 with probability 0.6 (and 0 with prob. 0.4). Consider lottery A, which gives you $1,000 with probability 2/3 (and 0 with prob. 1/3), and lottery B, which gives you a 50-50 chance of $300 and $700. Which lottery do you prefer if you are an expected utility maximizer?