0.5 Quadratic equations

The term D can be expressed as

We can tell much about the nature of the plot of D by examining this equation. D is a parabola with a vertex at (-1/2, -12.25). The parabola would be concave positive. The parabola would cross the n -axis at the values of the roots, namely -4 and 3. Figure 2 presents a sketch of D as a function of n .

This sketch presents us with invaluable information with regard to weighing the costs associated with the two manufacturing processes. To begin, we see that the value of D is positive for values of n >3. We interpret this to mean that whenever the number of aircraft produced in a 30-day production run exceeds the number 3, it is more cost efficient to implement manufacturing process B. Alternatively, whenever the number of aircraft produced in a 30-day production run is greater than or equal to 0 and less than 3, manufacturing process A is the most cost efficient. If 3 aircraft are manufactured in a 30-day production run, the costs associated with process A and process B are equal.

Summary

A thorough knowledge of quadratic equations is crucial for an engineer. This module has presented several applications of quadratic equations in the context of problems that occur in the study of engineering. We see that the motion of dragsters, the flight of rockets and determining the most cost-effective approach to competing manufacturing processes involve quadratic polynomials and equations.

Exercises

1. Wind power is currently being considered as a primary source for generating electricity. The pressure due to the wind ( P : measured in lbs/ft ² ) is related to the velocity of the wind ( v : measured in miles/hr) by the equation

Determine the approximate wind velocity ( v ) if the pressure is measured to be 11.75 lb/ft ² .

2. The distance ( d : measured in miles) to the apparent visible horizon of an observer at a height ( h : measured in miles) above sea level is given by the formula

On a clear day, how far away is the visible horizon for a passenger flying aboard an aircraft at a height of 6 miles?

3. The braking distance ( d : measured in feet) for a car travelling at a velocity = v miles/hour with good tires and well maintained brakes, on dry pavement can be calculated by the formula, $d=\frac{v(v+\text{20})}{\text{20}}$ . If a child runs into the street 100 feet in front of your car and you react immediately, what is the maximum speed that you could be driving and still stop without hitting the child?

4. An industrial engineer using supply chain management techniques, estimates that the hourly cost (C measured in dollars/hr) of producing n sub-assemblies for a robotic application by the formula, $C=2n^2+\text{50}n-\text{30}$

How many subassemblies can be produced when C = $270/hr?

5. The Little Old Lady from Pasadena is a very cautious driver who is on her way to the grocery store. The speedometer on her car registers an initial velocity of of 25 miles/hr. She presses on her accelerator to produce a constant acceleration of 3,600 miles/hr ² . Assuming that she is 10 miles from her final destination when she begins to accelerate, how long will it for her to reach the grocery store?

6. A mortar shell is launched at time, t = 0, with an initial velocity of 10,000 m/s. When will the shell hit the Earth?

7. Sketch the height of the mortar shell presented in exercise 6 as a function of time.

8. A mortar shell is launched from a bluff overlooking a battlefield. The height of the bluff is 250 m above the target of the shell. Assuming that the initial velocity of the shell is 5,000 m/s, how long will it take for the shell to travel to its target?

9. The expected profit from the manufacturing of a particular item is governed by a quadratic equation $P(n)=-n^2+\text{120}n$ . In the equation, n represents the number of items manufactured. Sketch the curve of P ( n ).

10. Using the information given in exercise 9, what is the value of the number of items that should be manufactured to produce the maximum amount of profit?