question 47 0/1 pt 2 98 details if a projectile is fired with a velocity of v0 at an angle θ, then its…

question 47 0/1 pt 2 98 details if a projectile is fired with a velocity of v0 at an angle θ, then its range, the horizontal distance it travels (in ft), is modeled by the function r(θ)=(v0)²sin(2θ)/32. if v0 = 360 feet per second, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 2460 feet away? (round your answer to the nearest hundredth.) question help: video message instructor post to forum submit question jump to answer

question 47 0/1 pt 2 98 details if a projectile is fired with a velocity of v0 at an angle θ, then its range, the horizontal distance it travels (in ft), is modeled by the function r(θ)=(v0)²sin(2θ)/32. if v0 = 360 feet per second, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 2460 feet away? (round your answer to the nearest hundredth.) question help: video message instructor post to forum submit question jump to answer

Answer

Explanation:

Step1: Substitute given values into formula

Given $v_0 = 360$, $R(\theta)=2460$, and the formula $R(\theta)=\frac{(v_0)^2\sin(2\theta)}{32}$. Substitute to get $2460=\frac{(360)^2\sin(2\theta)}{32}$.

Step2: Solve for $\sin(2\theta)$

First, simplify the right - hand side of the equation. $\frac{(360)^2}{32}=\frac{129600}{32} = 4050$. So the equation becomes $2460 = 4050\sin(2\theta)$. Then $\sin(2\theta)=\frac{2460}{4050}=\frac{82}{135}$.

Step3: Find $2\theta$

Using the inverse - sine function, $2\theta=\sin^{-1}(\frac{82}{135})$. Calculate $\sin^{-1}(\frac{82}{135})\approx37.79^{\circ}+360^{\circ}n$ or $180^{\circ}- 37.79^{\circ}+360^{\circ}n$, $n\in\mathbb{Z}$. Since we are looking for angles in the range of a projectile motion ($0^{\circ}\leq\theta\leq90^{\circ}$), we consider the principal value. So $2\theta\approx37.79^{\circ}$.

Step4: Solve for $\theta$

Divide both sides of $2\theta\approx37.79^{\circ}$ by 2. $\theta=\frac{37.79^{\circ}}{2}\approx18.90^{\circ}$.

Answer:

$18.90$