If a projectile is fired with velocity v0 at an angle θ, then its range, the horizontal distance it travels (in feet), is modeled by the function
R(θ) =
v02 sin 2θ
32
.
If
v0 = 2600 ft/s,
what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft away? (Enter your answers as a comma-separated list. Round your answers to three decimal places.)
Thanks
To find the angle (θ) at which the projectile should be fired, we can set the range equation equal to the given distance:
R(θ) = 5000 ft
Substituting the given values into the equation for R(θ), we have:
5000 = (2600^2 * sin(2θ))/(32)
To solve for θ, we need to isolate sin(2θ). Multiply both sides of the equation by 32:
160000 = 2600^2 * sin(2θ)
Divide both sides of the equation by 2600^2:
(160000) / (2600^2) = sin(2θ)
Take the inverse sin of both sides:
sin^(-1) [(160000) / (2600^2)] = 2θ
Divide both sides of the equation by 2:
sin^(-1) [(160000) / (2600^2)] / 2 = θ
Using a calculator, evaluate sin^(-1) [(160000) / (2600^2)] / 2 to find the angle in radians. Finally, convert the angle from radians to degrees.
θ ≈ <<14.832>>14.832° (rounded to three decimal places)
Therefore, the angle at which the projectile should be fired is approximately 14.832°.
To find the angle (in degrees) at which the projectile should be fired in order to hit a target 5000 ft away, we can use the given range equation:
R(θ) = (v₀² * sin(2θ)) / 32
Given that v₀ = 2600 ft/s and the desired range is 5000 ft, we can substitute these values into the equation and solve for θ.
5000 = (2600² * sin(2θ)) / 32
Multiplying both sides by 32 and divding by (2600²), we get:
160000 = sin(2θ)
To find θ, we need to take the inverse sine (or arcsine) of both sides:
sin^(-1)(160000) = 2θ
Using a calculator to evaluate sin^(-1)(160000), we get:
179.212 = 2θ
Dividing both sides by 2, we get:
θ = 179.212 / 2
θ ≈ 89.606
Therefore, the angle should be approximately 89.606 degrees in order to hit the target 5000 ft away.
ind the key numbers of the expression. (Enter your answers as a comma-separated list.)
3x2 − x − 10