Math / Trigonometry

QuestionAn aircraft is climbing at a 30 degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is $500 \mathrm{mi} / \mathrm{hr}$ ? Express your answer as $\mathrm{mi} / \mathrm{hr}$ and round it into the nearest whole number. Input only the numerical value of the answer without the unit.

Studdy Solution

STEP 1

1. The aircraft is climbing at a constant angle of $30^\circ$ to the horizontal.
2. The speed of the aircraft is $500$ miles per hour.

STEP 2

1. Understand the relationship between the speed of the aircraft and the vertical component of the speed.
2. Use trigonometry to find the vertical component of the speed.
3. Calculate the vertical speed and round it to the nearest whole number.

STEP 3

Understand that the vertical component of the aircraft's speed can be found using the sine function, as the aircraft is climbing at an angle to the horizontal.

STEP 4

Use the sine function to find the vertical component of the speed:
$\text{Vertical Speed} = \text{Total Speed} \times \sin(\text{Angle})$
Substitute the given values:
$\text{Vertical Speed} = 500 \, \text{mi/hr} \times \sin(30^\circ)$

STEP 5

Calculate the sine of $30^\circ$ , which is $0.5$ , and then find the vertical speed:
$\text{Vertical Speed} = 500 \, \text{mi/hr} \times 0.5$ $= 250 \, \text{mi/hr}$
Round the result to the nearest whole number (though it is already a whole number).
The aircraft is gaining altitude at a speed of:
$\boxed{250}$

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