Quadratic Modeling Worksheets Grade 10

1. Noah throws a ball upward from a height of 8 meters with an initial velocity of 24 m/s. The height h (in meters) after t seconds is given by h(t) = -5t² + 24t + 8. Find the maximum height reached by the ball.
2. Mason throws a ball upward from a height of 2 meters with an initial velocity of 27 m/s. The height h (in meters) after t seconds is given by h(t) = -5t² + 27t + 2. Find the maximum height reached by the ball.
3. Mason is a farmer who wants to build a rectangular enclosure for his sheep against a straight riverbank. He has 72 meters of fencing. The side along the river does not need fencing. The width of the enclosure (the sides perpendicular to the river) is x meters, and the length (parallel to the river) is y meters. Mason wants to maximize the area of the enclosure. What width x (in meters) will give the maximum area?

…and 4 more problems

1. Matiu is a farmer who grows organic strawberries. He notices that the number of strawberry plants he can sell each week, n, depends on the price per plant, p dollars, according to the relationship n = 2400 - 60p. The cost to grow and package each plant is $8, and he has fixed weekly costs of $3200. Determine the price Matiu should charge per plant to maximize his weekly profit, and what is that maximum weekly profit?
2. A projectile is launched from ground level. Its height h (in meters) at time t (in seconds) is modeled by the quadratic function h(t) = -5t² + 30t. What is the maximum height reached by the projectile?
3. Isabella is designing a rectangular garden next to a straight river. She has 72 meters of fencing to enclose the other three sides of the garden (the side along the river needs no fencing). She wants to maximize the area of the garden. What dimensions (length parallel to the river and width perpendicular to the river) will give the maximum area, and what is that maximum area?

…and 4 more problems

1. A company is designing a new smartphone case and wants to model the relationship between the selling price and weekly profit. Their market research shows that when they sell the case for $20, they make a weekly profit of $800, and when they sell it for $30, they make a weekly profit of $1200. The maximum profit occurs at a selling price of $35. Find the quadratic function P(x) = ax² + bx + c that models their weekly profit, where x is the selling price in dollars and P(x) is the profit in dollars.
2. Hana is designing a parabolic water fountain for a community garden. The water jet follows a parabolic path, with the nozzle located at ground level at the origin (0, 0). The water reaches its maximum height of 8 meters at a horizontal distance of 6 meters from the nozzle. The water then lands back on the ground at a point (12, 0). On a coordinate grid, each unit represents 1 meter. Hana wants to place a decorative ring at a height of 6 meters above the ground. How far horizontally from the nozzle will the water pass through this ring?
3. Mere is designing a parabolic water fountain for a park. The fountain's water stream follows a parabolic path. The water spout is at ground level, and the stream reaches a maximum height of 4 meters at a horizontal distance of 6 meters from the spout. The stream lands back on the ground at a horizontal distance of 12 meters from the spout. Mere wants to know the height of the water stream at a horizontal distance of 4 meters from the spout. What is this height?

…and 3 more problems