A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through a
n angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?
1 answer:
Answer:
θ = 60°
Step-by-step explanation:
The cross sectional area of the trapezoid shape will be that of a trapezoid with bases of 10 cm and (10 cm + 2·(10 cm)·cos(θ)) and height (10 cm)·sin(θ).
That area in cm² is ...
A = (1/2)(b1 +b2)h = (1/2)(10 + (10 +20cos(θ))(10sin(θ)
A = 100sin(θ)(1 +cos(θ))
A graphing calculator shows this area to be maximized when ...
θ = π/3 radians = 60°
_____
<em>A</em> will be maximized when its derivative with respect to θ is zero. That derivative can be found to be 2cos(θ)² +cos(θ) -1, so the solution reduces to ...
cos(θ) = 1/2
θ = arccos(1/2) = π/3
You might be interested in
Hi there!
Let's solve this equation step by step!
6f + 9g = 3g + f
To solve for f, we need to bring all the terms in the equation with an f in it to the left, and all the other terms (with a g) to the right.
First subtract f from both sides.
5f + 9g = 3g
Now subtract 9g from both sides.
5f = -6g
And finally divide both sides by 5.
f = (-6/5)g
Hence, your answer;
~ Hope this helps you!
Let's solve this equation step by step!
6f + 9g = 3g + f
To solve for f, we need to bring all the terms in the equation with an f in it to the left, and all the other terms (with a g) to the right.
First subtract f from both sides.
5f + 9g = 3g
Now subtract 9g from both sides.
5f = -6g
And finally divide both sides by 5.
f = (-6/5)g
Hence, your answer;
~ Hope this helps you!
So I'm going to assume that this question is asking for <u>non extraneous solutions</u>, or solutions that are found in the equation <em>and</em> are valid solutions when plugged back into the equation. So firstly, subtract 2 on both sides of the equation:
Next, square both sides:
Next, subtract x and add 2 to both sides of the equation:
Now we are going to be factoring by grouping to find the solution(s). Firstly, what two terms have a product of 6x^2 and a sum of -5x? That would be -3x and -2x. Replace -5x with -2x - 3x:
Next, factor x^2 - 2x and -3x + 6 separately. Make sure that they have the same quantity on the inside of the parentheses:
Now you can rewrite the equation as
Now, apply the Zero Product Property and solve for x as such:
Now, it may appear that the answer is C, however we need to plug the numbers back into the original equation to see if they are true as such:
Since both solutions hold true when x = 2 and x = 3, <u>your answer is C. x = 2 or x = 3.</u>