Quadratic Formula Word Problems Worksheet

A quadratic formula word problems worksheet is a valuable tool for students who want to improve their understanding and application of quadratic equations. This type of worksheet provides a variety of real-world scenarios that require the use of the quadratic formula to solve. By presenting these problems in a structured format, students can practice identifying the key elements of each problem, such as the entity and subject, and then applying the quadratic formula to find the solution.

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What is the quadratic formula and how is it used to solve quadratic equations?

The quadratic formula is: x = (-b ± ?(b² - 4ac)) / 2a. It is used to find the solutions (or roots) of a quadratic equation of the form ax² + bx + c = 0. By plugging the coefficients a, b, and c into the formula, we can calculate the values of x that satisfy the equation. The ± symbol indicates that there are usually two potential solutions for a quadratic equation.

A ball is thrown into the air with an initial velocity of 25 m/s. How long does it take for the ball to hit the ground?

Assuming the ball is thrown upwards and then falls back down, we can calculate the time it takes for the ball to hit the ground by considering the vertical motion. Using the kinematic equation y = y0 + v0t + 0.5at^2, where y is the final position (0 since it hits the ground), y0 is the initial position (0), v0 is the initial velocity (25 m/s), a is the acceleration due to gravity (-9.8 m/s^2 downwards), and t is the time, we can solve for t. Plugging the values into the equation, we get 0 = 0 + 25t + 0.5(-9.8)t^2. Solving for t, we find that it takes approximately 5.1 seconds for the ball to hit the ground.

The profit function of a company is given by P(x) = -2x^2 + 50x - 100, where x represents the number of products sold. How many products must be sold to maximize profit?

The profit function P(x) is a quadratic function that represents a parabolic shape. To find the number of products that must be sold to maximize profit, we can find the vertex of the parabola using the formula x = -b/(2a) where a = -2 and b = 50. Plugging in these values, we get x = -50/(2*-2) = 12.5. Since the number of products must be a whole number, the company must sell 13 products to maximize profit.

A farmer wants to create a rectangular pen with an area of 400 square meters. If the width of the pen is 10 meters shorter than its length, what are the dimensions of the pen?

Let the length of the pen be x meters. Since the width is 10 meters shorter, the width would be x - 10 meters. The area of a rectangle is given by length times width, so we have the equation x(x - 10) = 400. Expanding and rearranging, we get x^2 - 10x - 400 = 0. Solving this quadratic equation, we find that x = 20 meters (length) and x - 10 = 10 meters (width). Therefore, the dimensions of the pen are 20 meters in length and 10 meters in width.

The height of a triangle is 10 cm less than twice its base length. If the area of the triangle is 48 cm², what are the dimensions of the triangle?

Let the base length of the triangle be x cm. Therefore, the height of the triangle is 2x - 10 cm. Using the formula for the area of a triangle (Area = 1/2 * base * height), and substituting the given values, we get: 48 = 1/2 * x * (2x - 10). Simplifying this equation, we find x² - 5x - 48 = 0. By factoring, we obtain (x - 8)(x + 3) = 0. So, x = 8 or x = -3. Since the base length cannot be negative, the base length is 8 cm. Substituting back into the height equation gives the height as 2(8) - 10 = 16 - 10 = 6 cm. Therefore, the dimensions of the triangle are a base of 8 cm and a height of 6 cm.

A car rental company charges $30 per day plus $0.20 per mile driven. If the total cost for renting a car for a day is $60, how many miles were driven?

To calculate the number of miles driven, we can start by subtracting the fixed daily charge of $30 from the total cost of $60, leaving us with $30. This remaining amount represents the cost incurred from the miles driven. Since the company charges $0.20 per mile, we can divide $30 by $0.20 per mile to find that 150 miles were driven in total for the day.

The profit function for a certain product is given by P(x) = -3x^2 + 120x - 500, where x represents the number of items produced. What is the maximum profit that can be earned?

The maximum profit that can be earned is $800 when x = 20.

The length of a rectangle is 5 meters longer than its width. If the perimeter of the rectangle is 40 meters, what are its dimensions?

The width of the rectangle is 5 meters, and the length is 10 meters.

The height of a trapezoid is 8 cm. The shorter base is twice the length of the longer base, and the area of the trapezoid is 120 cm². Find the lengths of the bases.

Let the longer base be x cm, then the shorter base is 2x cm. Using the formula for the area of a trapezoid (area = 1/2*height*(sum of bases)), we get 120 = 1/2*8*(x + 2x) = 1/2*8*3x = 12x. Solving for x, x = 10 cm. Therefore, the longer base is 10 cm and the shorter base is 20 cm.

A rock is thrown vertically upward from ground level with an initial velocity of 25 m/s. How high does the rock go?

The maximum height the rock reaches can be calculated using the kinematic equation: \(h = \frac{v^2 - u^2}{2g}\), where \(h\) is the maximum height, \(v\) is the final velocity (0 m/s at the peak), \(u\) is the initial velocity (25 m/s), and \(g\) is the acceleration due to gravity (9.81 m/s²). Plugging in these values, we find that the rock reaches a height of approximately 31.9 meters.