A hot-air balloon is rising vertically 5 m/s while the wind is blowing horizontally at 3 m/s. Find the speed v of the balloon and the angle 0 it makes with the horizontal. 5 m/s Balloon 3 m/s Wind

Identify components: Identify the components of the balloon's velocity. The balloon is rising vertically at 5 m/s and the wind is blowing horizontally at 3 m/s.Use Pythagorean theorem: Use the Pythagorean theorem to find the resultant speed v of the balloon. The vertical and horizontal components of the velocity are perpendicular to each other, so we can treat them as the legs of a right triangle, with the balloon's speed v as the hypotenuse. v^2 = (vertical speed)^2 + (horizontal speed)^2 v^2 = 5^2 + 3^2 v^2 = 25 + 9 v^2 = 34 v = sqrt(34) v ≈ 5.83 m/sCalculate resultant speed: Calculate the angle θ the balloon's path makes with the horizontal. We can use the tangent function, which is the ratio of the opposite side (vertical speed) to the adjacent side (horizontal speed) in a right triangle. tan(θ) = (vertical speed) / (horizontal speed) tan(θ) = 5 / 3 To find θ, we take the arctan (inverse tangent) of 5/3. θ = arctan(5/3) θ ≈ 59.04 degrees

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A hot-air balloon is rising vertically $5\,\text{m/s}$ while the wind is blowing horizontally at $3\,\text{m/s}$ . Find the speed $v$ of the balloon and the angle $\theta$ it makes with the horizontal. $5\,\text{m/s}$ Balloon $3\,\text{m/s}$ Wind

View step-by-step help

Home

Math Problems

Algebra 2

Pythagorean Theorem and its converse

Full solution

Q. A hot-air balloon is rising vertically

5\,\text{m/s}

while the wind is blowing horizontally at

3\,\text{m/s}

. Find the speed

v

of the balloon and the angle

\theta

it makes with the horizontal.

5\,\text{m/s}

Balloon

3\,\text{m/s}

Wind

Identify components: Identify the components of the balloon's velocity. $\newline$ The balloon is rising vertically at $5\,\text{m/s}$ and the wind is blowing horizontally at $3\,\text{m/s}$ .
Use Pythagorean theorem: Use the Pythagorean theorem to find the resultant speed $v$ of the balloon. $\newline$ The vertical and horizontal components of the velocity are perpendicular to each other, so we can treat them as the legs of a right triangle, with the balloon's speed $v$ as the hypotenuse. $\newline$ $v^2 = (\text{vertical speed})^2 + (\text{horizontal speed})^2$ $\newline$ $v^2 = 5^2 + 3^2$ $\newline$ $v^2 = 25 + 9$ $\newline$ $v^2 = 34$ $\newline$ $v = \sqrt{34}$ $\newline$ $v \approx 5.83$ m/s
Calculate resultant speed: Calculate the angle $\theta$ the balloon's path makes with the horizontal. $\newline$ We can use the tangent function, which is the ratio of the opposite side (vertical speed) to the adjacent side (horizontal speed) in a right triangle. $\newline$ $\tan(\theta) = \frac{\text{vertical speed}}{\text{horizontal speed}}$ $\newline$ $\tan(\theta) = \frac{5}{3}$ $\newline$ To find $\theta$ , we take the arctan (inverse tangent) of $\frac{5}{3}$ . $\newline$ $\theta = \arctan\left(\frac{5}{3}\right)$ $\newline$ $\theta \approx 59.04$ degrees