Answer:

14.27

Step-by-step explanation:

To find out how long it will take for the plane to climb to an altitude of 1000 meters, you can use trigonometry. First, you need to determine the vertical component of the plane's velocity.

Given:

- The plane's take-off angle to the horizontal, θ = 32 degrees.

- The plane's horizontal velocity, v = 180 km/hr.

You want to find the time it takes to climb to an altitude of 1000 meters (h = 1000 m).

The vertical component of velocity (v_vertical) can be found using trigonometry:

v_vertical = v * sin(θ)

Convert the speed from km/hr to m/s (since 1 km/hr = 1000/3600 m/s):

v = 180 km/hr = (180 * 1000 m) / 3600 s = 50 m/s

Now, calculate the vertical velocity component:

v_vertical = 50 m/s * sin(32 degrees)

Next, you can use the following kinematic equation to find the time (t) it takes to reach an altitude of 1000 meters:

h = 0.5 * g * t^2

Where:

- h = 1000 m (the altitude)

- g = 9.81 m/s^2 (acceleration due to gravity)

Rearrange the equation to solve for time (t):

t = √(2h / g)

Now, plug in the values:

t = √(2 * 1000 m / 9.81 m/s^2)

Calculate the time:

t ≈ √(203.57) ≈ 14.27 seconds

So, it will take approximately 14.27 seconds for the plane to climb to an altitude of 1000 meters.

Verified answer

Answer:

0.01048hrs

Step-by-step explanation:

To solve this problem, we can use trigonometry. The altitude (height) the plane reaches is the opposite side of the right triangle formed with the horizontal, and the angle of ascent is the angle between the hypotenuse (the path of the plane) and the horizontal.

We can use the sine function to relate the angle, the hypotenuse, and the opposite side:

[tex]\[ \sin(\theta) = \dfrac{\text{opposite}}{\text{hypotenuse}} \][/tex]

Now, we can solve for the hypotenuse (the distance the plane travels along its path) using the given angle and the altitude:

[tex]\[ \sin(32^\circ) = \dfrac{1000\, \text{m}}{\text{hypotenuse}} \][/tex]

Now solve for the hypotenuse:

[tex]\[ \text{hypotenuse} = \dfrac{1000\, \text{m}}{\sin(32^\circ)} = 1887.07 \][/tex]

Now, we can use the formula for speed to find the time it takes to travel this distance:

[tex]\[ \text{speed} = \dfrac{\text{distance}}{\text{time}} \][/tex]

We need to convert the speed from km/hr to m/s to match the units of the distance:

[tex]\[ \text{speed} = 180\, \text{km/hr} \cdot \left(\dfrac{1000\, \text{m}}{1\, \text{km}}\right) \cdot \left(\dfrac{1\, \text{hr}}{3600\, \text{s}}\right) = 50\, \text{m/s} \][/tex]

Now, solve for time:

[tex]\[ 50\, \text{m/s} = \dfrac{\text{hypotenuse}}{\text{time}} \][/tex]

[tex]\[ \text{time} = \dfrac{\text{hypotenuse}}{50\, \text{m/s}} \][/tex]

[tex]\[ \text{time} = \dfrac{\text{1887.07}}{50\, \text{m/s}} \][/tex]

[tex]\[ \text{time} = 37.7414 \: sec≈0.01048hrs\][/tex]

[tex] \rule{180pt}{3pt} [/tex]