Section 1 of 13
Aerospace engineers design primarily aircraft spacecraft, satellites, and missiles.
Entry-level aerospace engineers usually need a bachelor’s degree. Bachelor’s degree programs include classroom, laboratory, and field studies in subjects such as general engineering principles, propulsion, stability and control, structures, mechanics, and aerodynamics, which is the study of how air interacts with moving objects.
Section 3 of 13
Section 4 of 13
In this lesson, you will look at how an aerospace engineer uses quadratic functions to model the path of a flight or the forces acting on an aircraft.
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Section 5 of 13
Section 6 of 13
When a spacecraft travels beyond Earth’s atmosphere, the gravitational force is close to zero. To simulate microgravity on Earth, aerospace engineers use a special type of aircraft. The pilot flies the aircraft upward and then cuts the thrust, allowing the aircraft to continue rising and then fall back down in a parabolic pattern before increasing the thrust again.
The graph shows the height in meters of the aircraft as a function of time in seconds after the pilot begins to fly upward. The height is measured relative from when the pilot cuts the thrust. The solid line indicates the part of the flight that is parabolic, when the gravity is near zero. Identify the vertex and x-intercepts of the parabola, and interpret what each represents in this situation. Then write a quadratic function in intercept form for the parabola.
Identify and interpret the vertex.
Identify and interpret the x-intercepts of the parabola. The vertex is (30, 800). It represents the maximum height of the jet, 800 meters above the point where the pilot cut the thrust. The jet reaches that point 30 seconds after the beginning of the maneuver.
Identify and interpret the x-intercepts of the parabola.
The x-intercepts are 20 and 40. They represent the beginning and ending times of the parabolic portion of the flight. The pilot cuts the thrust 20 seconds into the maneuver and increases the thrust again at 40 seconds.
Write the function for the parabola.
To write a function for the parabola in intercept form, substitute 20 and 40 for p and q in
f(x) = (x − p)(x − q).
f(x)
= a(x − p)(x − q)
= a(x − 20)(x − 40)
Then use the vertex, (30, 800), to solve for a.
800 = a(30 − 20)(30 − 40)
800 = a(10)(−10)
800 = −100a
−8 = a
The function in intercept form for the parabola is
f(x) = −8(x − 20)(x − 40).
Answers: x-intercepts: 0, 20; vertex: (10, 800); f(x)=−8x(x− 20)
Section 7 of 13
Section 8 of 13
An aerospace engineer represents the height in meters of an aircraft during a parabolic maneuver, relative to the height when the pilot reduces the thrust, using the function f of x equals negative 8 times open paren x minus 20 close paren times open paren x minus 40 close paren, where x is the time in seconds. Suppose the pilot cuts the thrust when the aircraft is at an altitude of 9000 meters. The engineer writes a new function where y represents the altitude of the aircraft. What is the function in
intercept form for the altitude of the jet that the engineer writes? Then graph the new function.
Write a function for the altitude of the aircraft.
For any height given by the original function, the altitude of the aircraft is 9000 meters greater. So, the altitude is given by the function f(x) = -8(x – 20)(x – 40) + 9000.
To rewrite the function in intercept form, first simplify and then factor.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
Graph the function.
To graph the function, first plot the intercepts at (−5, 0) and (65, 0). Then locate the line of symmetry halfway between the intercepts, x = 30. Substitute 30 for x into the function to find the y-coordinate of the vertex.
f(x) = −8(x + 5)(x − 65)
= −8(30 + 5)(30 − 65)
= −8(35)(−35)
= 9800
Plot the vertex at (30, 9800). Use the vertex,
intercepts, and line of symmetry to draw the
parabola.
Section 9 of 13
Use this formula sheet if needed.
Section 10 of 13
In order to design aircraft, aerospace engineers need to
understand the forces that act on an object as it moves through the air. Thrust is the force that moves an airplane forward. Drag is
the force that works against thrust, slowing the airplane down.
An aerospace engineer writes the function
f(x) = 0.15×2 – 27x + 1500
for the drag y, in pounds, of an aircraft as a function of airspeed x,
in knots. At which airspeed is there the least amount of drag?
Graph the function.
To graph the function, first plot the vertex at (90, 285). Then substitute a value for x to find another point on the graph.
f(x) = 0.15×2 − 27x + 1500
= 0.15(x2 − 180x) + 1500
= 0.15(x2 − 180x + 8100) + 1500 − 0.15(8100)
= 0.15(x − 90)2 + 285
Rewrite the function in vertex form by completing the square.
f(x) = 0.15×2 − 27x + 1500
= 0.15(x2 − 180x) + 1500
= 0.15(x2 − 180x + 8100) + 1500 − 0.15(8100)
= 0.15(x − 90)2 + 285
= 660
The graph passes through the point (40, 660). Draw the left half of the parabola through (40, 660) and ending at the vertex. The line x = 90 is a line of symmetry, so draw the right half of the parabola to mirror the left.
Step 2 shows that the point (40, 660) lies on the graph. What other point on the graph has a y-coordinate of 660? How do you know?
Step 2 shows that the point (40, 660) lies on the graph. What other point on the graph has a y-coordinate of 660? How do you know?
Find the airspeed that produces the minimum amount of drag.
The vertex represents the minimum value of the function. The least amount of drag on the plane is 285 pounds, when the airspeed is 90 knots.
Answers:
(80, 285); The new graph is the original shifted left 10 units. When x= 30, y= 660
Section 11 of 13
Section 12 of 13
Section 13 of 13
Use this formula sheet if needed.