Section Intro to Solve Problems using Quadratic Equations

Finding Area Dimension from Given Conditions

ex Two squares whose side lengths are consecutive odd integers have a total area of 650 cm^2. What is their total perimeter?

Therefore, the total perimeter is 144 cm.

Finding Dimension of a Park

ex In laying out a park, the landscape artist includes a rectangular flowerbed 18 m by 12 m. Within the bed, she plans to have a uniform-width border of annuals around a permanent rose bed. To obtain the best appearance, she wants the rose bed to be exactly half the total area. What is the width of the border, correct to one decimal place?

Create a quadratic model for the height of a toy rocket launched upward at 45 m/s from a 2-m platform.

How long would the rocket take to fall to Earth, rounded to the nearest hundredth of a second?

A firework is launched upward at an initial velocity of 49 m/s, from a height of 1.5 m above the ground. The height of the firework, in metres, after t seconds, is modelled by the equation h=-4.9t^2+49t+1.5

a) What is the maximum height of the firework above the ground?

A firework is launched upward at an initial velocity of 49 m/s, from a height of 1.5 m above the ground. The height of the firework, in metres, after t seconds, is modelled by the equation h=-4.9t^2+49t+1.5

Over what time interval is the height of the firework greater than 100 m above the ground? Round to the nearest hundredth of a second.

The length of a rectangle is 16 m greater than its width. The area is 35m^2. Find the dimensions of the rectangle, to the nearest hundredth of a metre.

The product of two consecutive numbers is 3306. What are the numbers?

Determine two consecutive odd integers whose product is 323.

The length of one leg of a right triangle is 7 cm more than that of the other leg. The length of the hypotenuse is 3 cm more than double that of the shorter leg. Find the lengths of each of the three sides.

A cylindrical can with height 12 cm has capacity 600 mL. What is its radius. to the nearest millimetre? [Remember that 1 mL = 1 cm^3.]

The area of a triangle is 20 cm^2, and the altitude is 4 cm greater than the base. Find the length of the base. to the nearest millimetre.

The sum of the squares of two consecutive integers is 365. Find the integers.

A rectangle has perimeter 23 cm. Its area is 33 cm^2. Determine the dimensions of the rectangle. Include a diagram in your solution.

A rectangular construction site is enclosed on three sides using 1200 m of fencing. The remaining side is formed by an existing wall. What dimensions enclose 180 000 m^2 of land?

The three sides of a right triangle are consecutive even integers. What is the length of each side?

A ladder is 6 m long. If the height of the top of the ladder must be no greater than 10 times the distance from the base to the wall, how high up a wall can the top of the ladder be placed? Include a diagram in your solution. Round to the nearest millimetre.

A science experiment involves launching a small rocket. The following measurements are taken:

Initial height: 0.61 m

Initial vertical velocity: 36.85 m/s

a) Create a quadratic model for the height, in metres, of the rocket after a given number of seconds.

b) Verify the following results of the experiment:

Total time in the air: 7.54 s

Maximum height: 69.89 m

c) Sketch a graph of this relation and label the key information as in Example 1 of this section.

The acceleration due to gravity on Earth is 9.8 m/s^2. A ball is thrown upward at an initial velocity of 15 m/s from a height of 1 m above the ground. Round answers to the nearest tenth.

a) Write an equation for the height of the ball.

b) What is the height of the ball after 1 s?

c) After how many seconds does the ball land?

d) What is the maximum height of the ball? When does this occur?

Shelly sells photos of athletes to baseball, basketball, and hockey fans after their games. Her regular price is $10 per photograph, and she usually sells about 30 photographs. Shelly finds that, for each reduction in price of $0.50, she can sell an additional two photographs.

a) Total sales revenue is the product of the number of units sold and the price. Make an algebraic model to represent Sherri’s total sales revenue.

b) At what price will Sherri’s revenue be $150?

c) At what price will her maximum revenue occur?

d) At what price will her revenue be $0?

e) Graph the relationship between revenue and the number of price reductions. Which features on the graph represent the solutions to parts b), c), and d)?

A rectangular picture frame measures 20 cm by 30 cm. A new frame is to be made by increasing each side length by the same amount.

The resulting enclosed area is to be 1064 cm^2. Find the dimensions of the new picture frame. Include a diagram in your solution.

A rectangular garden measures 15 m by 24 m. A larger garden is to be made by increasing each side length by the same amount. The resulting area is to be 1.5 times the original area. Find the dimensions of the new garden, to the nearest tenth of a metre. Include a diagram in your solution.

The length of a rectangular field is 2 m greater than three times its width. The area of the field is 1496 m^2. What are the dimensions of the field?

An open-topped box is to be made from a rectangular piece of tin measuring 50 cm by 40 cm by cutting squares of equal size from each corner. The base area is to be 875 cm^2

a) Draw a diagram representing the information.

b) What is the side length of the squares being removed?

c) What is the volume of the box?

A photograph measures 21 cm by 15 cm. A strip of constant width is to be cut from each side of the photo. so the area is reduced to 216 cm^2. Find the width of the cut. Include a diagram in your solution.

A photograph measures 20 cm by 16 cm. A strip of constant width is to be cut off the top and one side of the photo, so the area is reduced to 60\% of the area of the original photo. Find the width of the cut. Include a diagram in your solution.

A rectangular field measures 15 m by 20 m. A rectangular area is to be fenced in by reducing each dimension by the same 1 amount. The fenced-in area will be \displaystyle{\frac{1}{2}} the original area. What will the dimensions of the fenced-in area be? Include a diagram in your solution.

A rotating liquid surface takes on the shape of a parabolic mirror. The diameter of the mirror is 6 m. The vertex is 23 cm below the edges. Find an equation to model the parabolic cross section of the mirror.

Determine the number of points of intersection of each pair of parabolas. Justify your answer.

• \displaystyle y = x^2 + 2x + 7
• \displaystyle y = x^2 -4x -1

Determine the number of points of intersection of each pair of parabolas. Justify your answer.

• \displaystyle y = 3x^2 - 12x + 16
• \displaystyle y = -2x^2 -4x + 3

Determine the number of points of intersection of each pair of parabolas. Justify your answer.

• \displaystyle y = x^2 - 6x + 10
• \displaystyle y = 5x^2 -30x + 46

In the TV show Junkyard Wars, a trebuchet was used to catapult a pumpkin from a height of 4 m for a total horizontal distance of 24 m. It reached a maximum height of 14 m. At what horizontal distances was the height of the pumpkin 10 m, to the nearest metre?