Math Problem Statement
Solution
Let’s analyze the image step-by-step to solve each part of the question.
(a) Derivation of the Expression for R(θ)
The given problem walks through the simplification of using trigonometric identities:
- Using Distributive Property: Expanding terms within parentheses.
- Using Double-Angle Formulas: Specifically, and .
- Further Simplification: Leads to the final simplified form.
(b) Finding the Angle θ that Maximizes R
To find the angle that maximizes , solve .
The solution shows:
This angle maximizes .
(c) Maximum Distance R when feet per second
Now, plug feet per second and into the equation for to find the maximum distance .
Let’s calculate this.
-
The formula becomes:
-
Substitute values and calculate , rounding to two decimal places as needed.
Would you like me to proceed with the detailed calculation for in part (c)?
Related Questions
- How would the value of change if the initial velocity were doubled?
- Why is the optimal angle for maximum range on the inclined plane?
- What trigonometric identities are most useful for simplifying expressions like ?
- How would air resistance affect the maximum range ?
- Can this method of finding the maximum distance be applied to non-inclined planes?
Tip:
Remember that maximizing a trigonometric function often involves using identities and derivatives for simplification.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Optimization
Double-Angle Formulas
Formulas
R(θ) = (v₀²√2 / 16) * cos(θ) * (sin(θ) - cos(θ))
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)
Theorems
Double-Angle Theorem
Suitable Grade Level
Grades 11-12
Related Recommendation
Optimal Launch Angle on an Inclined Plane for Maximum Distance
Projectile Motion: Relationship Between Trajectory Angle, Range, and Elevation
Projectile Motion: Mini-Monster Truck Jump on a Slope
Solve for Projectile Range Given Initial Velocity and Acceleration
Projectile Motion: Calculating Range on an Incline with Initial Velocity of 200 m/s