More Applications: Quadratics Day 3 Notes Our warm-­‐up today involved falling objects! Now, we will focus on problems in which objects are being catapulted or launched! There is a new formula: ℎ = ℎ! + 𝑣! 𝑡 − 16𝑡 ! h =____________ t =____________ -­‐16t2 =____________ h0 =____________ v0 =____________ Practice writing the equation: 1. Initial height—6 feet; velocity 80 feet per second. 2. Initial height—180 feet; velocity 20.2 feet per second. 3. Velocity 68.9 feet per second; initial height—2 feet. Applications: 4. A cannon mounted on the top of 256ft tall fortress launches a shot in the air at an initial velocity of 96ft/sec upward. a. Write the equation that models the situation. b. How high is the cannonball after 3 seconds? c. When does the cannonball hit the ground? 5. A pumpkin is dropped from the top of a bridge that is 50 feet high. a. Write the equation that models the situation. b. What is the maximum height of the pumpkin? c. How long will it take for the pumpkin to hit the ground? Round to the nearest tenth of a second. 6. Cole is using Mrs. Peters’ stapler as a staple gun. He drops the stapler from a height of 3 feet. Assuming Cole does not catch the stapler, when will it hit the ground? (Round to the nearest hundredth of a second.) 7. A ground level super-­‐slingshot launches a pumpkin upward at an initial velocity of 88ft/sec. At what points in time will the pumpkin have a height of 120ft? More Applications: Quadratics Day 3 Notes What if a problem does not give you the initial velocity? 8. A basketball left Kemba’s hands at a point 8 feet above the ground. The basketball reached the basket (a height of 10 feet) 2.5 seconds after it left Kemba’s hands. a. Use this information to determine the initial upward velocity of the basketball. b. Write a rule giving h as a function of t. 9. The opening of a cannon is 16 feet above the ground. Mrs. Peters, who is shot out of the cannon, reaches a height of 52 feet after about 2 seconds. World Record Human Cannonball. a. Determine the upward velocity of Mrs. Peters. Show all work/equations! b. Write a rule that relates Mrs. Peters’ height above the ground h at a time t seconds after the cannon is fired. c. If for some unfortunate reason, the net slipped to the ground at the firing of the cannon, when would Mrs. Peters hit the ground? (Use 2nd Trace, Zero to calculate the exact number of seconds to the nearest hundredth of a second.) d. What is the exact maximum height of Mrs. Peters after she is shot from the cannon? (Use 2nd Trace, Maximum to calculate the maximum height to the nearest tenth). 10. At the Punkin’ Chunkin’ contest each year pumpkins are catapulted to great lengths! One of the pumpkins was launched from a height of 12 feet. It reached a height of 1,949 feet after 13 seconds of being in the air! a. Use this information to determine the initial upward velocity of the pumpkin. b. Write a rule giving h as a function of t. More Applications: Quadratics Day 3 Notes c. Use the equation above to find how long it takes for the pumpkin to hit the ground. Round to the nearest tenth of a second. Other applications: 11. You are in charge of constructing a two-­‐tower suspension bridge over Lake Norman. You have planned that the curve of the main suspension cables can be modeled by the function 𝑦 = 0.004𝑥 ! − 𝑥 + 80 where y represents the height of the cable above the bridge surface and x represents distance along the bridge surface from one tower toward the other. The values of x and y are measured in feet. a. What is the approximate height (from the bridge surface) of each tower from which the cable is suspended? b. What is the shortest distance from the cable to the bridge surface and where does it occur? 12. The Chickfila Dwarf House restaurant in Georgia has a miniature door visitors can use to enter the establishment. The following equation represents the archway of the door: 𝑦 = −3𝑥 ! +
54𝑥 − 235. Y represents the height of the door, while x represents the width of the door—
both measured in feet. What is the width of the archway to the nearest tenth of a foot?