Modeling with Quadratic Equations

To the nearest foot, what is the width of the arch 8 ft above the water?

15

The greatest rectangular area that the farmer can enclose with 100 m of fencing is

625 m2

The distance it takes a truck to stop can be modeled by the function d = stopping distance in feet v = initial velocity in miles per hour f = a constant related to friction When the truck's initial velocity on dry pavement is 40 mph, its stopping distance is 138 ft. Determine the value of f, rounded to the nearest hundredth.

f = 0.39

The main cable of a suspension bridge forms a parabola, described by the equation y = a(x − h)2 + k. y =height in feet of the cable above the roadway x =horizontal distance in feet from the left bridge support a =a constant (h, k) =vertex of the parbola What is the vertex of the parbola?

(105, 7)

The main cable attaches to the right bridge support at the same height as it attaches to the left bridge support. What is the distance between the supports?

180 ft

Choose the quadratic model for the situation.

C

To the nearest tenth of a second, how long after the pebble falls will it hit the ground?

3.5

The graph shows the height (h), in feet, of a basketball t seconds after it is shot. Projectile motion formula: h(t) = -16t2 + vt + h0 v = initial vertical velocity of the ball in feet per second h0 = initial height of the ball in feet Complete the quadratic equation that models the situation

24

The rancher decides to make the width of the rectangle 40 ft. What is the area of the rectangle?

2400

Find the length and width of the greatest rectangular area that the farmer can enclose with 100 m of fencing.

25 and 25

The main cable attaches to the left bridge support at a height of

26.25 ft

A rancher has a roll of fencing to enclose a rectangular area. The table shows how the area that the rancher can enclose with the fencing depends on the width of the rectangle. Which quadratic equation gives the area A of the rectangle in square feet given its width in w feet?

B

A farmer has 100 m of fencing to enclose a rectangular pen. Which quadratic equation gives the area (A) of the pen, given its width (w)?

C

A stone arch in a bridge forms a parabola described by the equation y = a(x - h)2 + k, where y is the height in feet of the arch above the water, x is the horizontal distance from the left end of the arch, a is a constant, and (h, k) is the vertex of the parabola. What is the equation that describes the parabola formed by the arch?

C

The function h(t) = -4.9t2 + h0 gives the height (h), in meters, of an object t seconds after it falls from an initial height (h0). The table shows data for a pebble that fell from a cliff. Choose the quadratic equation that models the situation.

C

The main cable of a suspension bridge forms a parabola, described by the equationy = a(x - h)2 + k, where y is the height in feet of the cable above the roadway, x is the horizontal distance in feet from the left bridge support, a is a constant, and (h, k) is the vertex of the parabola. At a horizontal distance of 30 ft, the cable is 15 ft above the roadway. The lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support. Which quadratic equation models the situation correctly?

C

Which quadratic equation models the situation correctly?

D

The quadratic function y = -10x2 + 160x - 430 models a store's daily profit (y), in dollars, for selling T-shirts priced at x dollars. A the daily profit the company would make from the T-shirts if it gave the T-shirts away for free B the greatest daily profit the company could make from selling the T-shirts C a selling price that would result in the company making no profit from the T-shirts D the selling price that would result in the company making the greatest daily profit from the T-shirts Match each item with what it represents in this situation by entering the appropriate letter in each box.

D B C A

The quadratic function y = -10x2 + 160x - 430 models a store's daily profit (y) for selling a T-shirt priced at x dollars. What equation do you need to solve to find the selling price or prices that would generate $50 in daily profit? What method would you use to solve the equation? Justify your choice.

Replace y in the equation with 50: (50 = -10x2 +160x - 430) Use factoring to solve the equation. After writing the equation in standard form and dividing each side by -10, it is easy to factor as 0 = (x - 4)(x - 12).