Question

FUS After takeoff, an aircraft enters a straight-line flight path with constant speed at point P(1.5|9|0.5). speed. In one minute it travels a distance vector v=(-1 5 0.2) back. Information in km. a) (1) Determine the speed of the aircraft in km/h. (2) After 4 minutes the plane reaches a big city. Calculate the altitude at this time. b) At an altitude of 2500 m the pilot switches to autopilot. Calculate the coordinates of the point where the aircraft is at this moment and state how much time has passed since the aircraft was at point P. c) Between the points A(-5|38|0.1), B(-4|40|0.11) and C (-7|39|0.1) there is a small natural protected area in the shape of a triangle. (1) Examine triangle ABC for special features. (2) Check whether the aircraft is flying over route AB and, if so, at what distance

60

likes
301 views

Answer to a math question FUS After takeoff, an aircraft enters a straight-line flight path with constant speed at point P(1.5|9|0.5). speed. In one minute it travels a distance vector v=(-1 5 0.2) back. Information in km. a) (1) Determine the speed of the aircraft in km/h. (2) After 4 minutes the plane reaches a big city. Calculate the altitude at this time. b) At an altitude of 2500 m the pilot switches to autopilot. Calculate the coordinates of the point where the aircraft is at this moment and state how much time has passed since the aircraft was at point P. c) Between the points A(-5|38|0.1), B(-4|40|0.11) and C (-7|39|0.1) there is a small natural protected area in the shape of a triangle. (1) Examine triangle ABC for special features. (2) Check whether the aircraft is flying over route AB and, if so, at what distance

Expert avatar
Santino
4.5
112 Answers
a)
(1) Zur Bestimmung der Geschwindigkeit des Flugzeugs in km/h teilen wir die zurückgelegte Strecke durch die Zeit:
\text{Geschwindigkeit } = \frac{\text{Strecke}}{\text{Zeit}} = \frac{\|\textbf{v}\|}{\text{Zeit}}

Gegeben: $\textbf{v} = (-1, 5, 0.2)$ km und Zeit = 1 Minute = 1/60 Stunde.

Berechnung der Geschwindigkeit:
\|\textbf{v}\| = \sqrt{(-1)^2 + 5^2 + 0.2^2} = \sqrt{1 + 25 + 0.04} = \sqrt{26.04} \approx 5.10 \text{ km}
\text{Geschwindigkeit} = \frac{5.10}{1/60} = 5.1 \times 60 = 306 \text{ km/h}

(2) Die Flughöhe nach 4 Minuten beträgt:
Höhe = 9 + (4 * 0.2) = 9.8 km

b)
Bei einer Höhe von 2500 m = 2.5 km schalten wir auf Autopilot um.

Die Koordinaten des Flugzeugs ergeben sich durch Addition des Vektors $\textbf{v}$ zum Punkt P:
(1 + (-1), 5 + 5, 0.5 + 0.2) = (0, 10, 0.7)

Die Zeit, die seit dem Punkt P vergangen ist, beträgt:
\frac{\sqrt{(0-1)^2 + (10-5)^2 + (0.7-0.5)^2}}{5.1} = \frac{\sqrt{1 + 25 + 0.04}}{5.1} \approx \frac{\sqrt{26.04}}{5.1} \approx \frac{5.1}{5.1} \approx 1 \text{ Stunde}

c)
(1)
Das Dreieck ABC ist ein gleichschenkliges Dreieck, da AB und BC jeweils die gleiche Länge haben.

(2)
Um zu prüfen, ob das Flugzeug die Strecke AB überfliegt, betrachten wir den Abstand des Flugzeugs von der Linie, die durch die Punkte A und B verläuft. Der Abstand kann durch den Normalenvektor auf die Ebene von A und B bestimmt werden.

Den Normalenvektor der Ebene bestimmen:
\textbf{n} = (A-B) \times (A-C) = \begin{pmatrix} -5+4 \ 38-40 \ 0.1-0.11 \end{pmatrix} \times \begin{pmatrix} -5+7 \ 38-39 \ 0.1-0.1 \end{pmatrix} = \begin{pmatrix} -1 \ -2 \ -0.01 \end{pmatrix} \times \begin{pmatrix} 2 \ -1 \ 0 \end{pmatrix}
= \begin{pmatrix} 0.01 \ -0.02 \ -3 \end{pmatrix}

Der Abstand des Flugzeugs von der Ebene beträgt:
\frac{| \textbf{n} \cdot (P - A) |}{\|\textbf{n}\|} = \frac{| \begin{pmatrix} 0.01 \ -0.02 \ -3 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 5 \ 0.5 \end{pmatrix} - \begin{pmatrix} 0.01 \ -0.02 \ -3 \end{pmatrix} \cdot \begin{pmatrix} -5 \ 38 \ 0.1 \end{pmatrix} |}{\sqrt{0.01^2 + (-0.02)^2 + (-3)^2}}

= \frac{| (0.01 \cdot 1 + (-0.02) \cdot 5 + (-3) \cdot 0.5) - (0.01 \cdot (-5) + (-0.02) \cdot 38 + (-3) \cdot 0.1) |}{\sqrt{0.01^2 + (-0.02)^2 + (-3)^2}}

= \frac{| 0.01 - 0.1 + 1.5 - 0.76 |}{\sqrt{0.01^2 + (-0.02)^2 + (-3)^2}} = \frac{| 2.67 |}{\sqrt{9.05}} \approx \frac{2.67}{3} \approx 0.89 \text{ km}

Antwort: a) (1) Die Geschwindigkeit des Flugzeugs beträgt 306 km/h.
b) Nach 4 Minuten beträgt die Flughöhe 9.8 km. Die Koordinaten des Flugzeugs sind (0, 10, 0.7) und es ist 1 Stunde vergangen, seit es sich im Punkt P befand.
c) (1) Das Dreieck ABC ist gleichschenklig.
(2) Das Flugzeug überfliegt die Strecke AB in einem Abstand von etwa 0.89 km.

Frequently asked questions (FAQs)
What is the period of the cotangent function, and what are its asymptotes?
+
What is 3/5 of 60 as a decimal?
+
What is the sum of 83 and 47?
+
New questions in Mathematics
How to find the value of x and y which satisfy both equations x-2y=24 and 8x-y=117
The sum of an infinite geometric series is 13,5 The sum of the same series, calculated from the third term is 1,5. Q. Calculate r if r>0.
Use the digits of 1,9,2,3 to come up with all the numbers 98 and 95
Karina has a plot of 5,000 square meters in which she has decided that 60% of it will be used to plant vegetables. Of this part, 12% will be dedicated to planting lettuce. How much surface area of the plot will be used for cultivation?
Given the vectors: a = (2m – 3n, 4n – m) and b = (2, -3), find the values of m and n that make: a = 5 b.
3(4x-1)-2(x+3)=7(x-1)+2
2.5 / 21.85
The actual length of an object is 1.3 m . If the blueprint uses a scale of 1 : 12 , what is the length of the line on the drawing?
Credit title that represents a payment order. This model, which emerged in Brazil, can only be issued in two specific situations: in the purchase and sale of commercial products or in the provision of services. Select the correct alternative: Question 6Answer The. Present value B. Promissory note w. Present value d. Duplicate It is. Bill of exchange
Shows two blocks, masses 4.3 kg and 5.4 kg, being pushed across a frictionless surface by a 22.5-N horizontal force applied to the 4.3-kg block. A. What is the acceleration of the blocks? B. What is the force of the 4.3-kg block on the 5.4 -kg block? C. What is the force of the 5.4 -kg block on the 4.3 -kg block?
How much does the average college student spend on food per month? A random sample of 50 college students showed a sample mean $670 with a standard deviation $80. Obtain the 95% confidence interval for the amount college students spend on food per month.
Show this compound proposition to be true or false. Paris is the capital of England or Rome is the capital of Italy
X³-27
For what values of m is point P (m, 1 - 2m) in the 2⁰ quadrant?
Sections of steel tube having an inside diameter of 9 inches, are filled with concrete to support the main floor girder in a building. If these posts are 12 feet long and there are 18 of them, how many cubic yards of concrete are required for the job?
8. Measurement Jillian measured the distance around a small fish pond to be 27 yards. What would be a good estimate of the distance across the pond: 14 yards, 9 yards, or 7 yards? Explain how you decided.
A membership to the gym cost $25 per person in 1995. The membership cost has increased by an average $6 per person for each year since 1995. Write a linear equation for the cost of a gym membership for one person since 1995. What is the cost of a gym membership in 2009?
How many moles are there in 235 grams of potassium thiosulfate pentahydrate? K2S2O3*5(H2O)
draw the condensed formula fpr 3,3,4 triethylnonane
Convert (324)𝑓𝑖𝑣𝑒 into base-ten