1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
UNO [17]
3 years ago
7

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? θ = radians
Mathematics
1 answer:
OverLord2011 [107]3 years ago
4 0

Answer:

  θ = π/3 radians

Step-by-step explanation:

The average width* of the gutter will be ...

  (10 cm)(1 + cos(θ))

and its depth will be ...

  (10 cm)sin(θ)

so the cross-sectional area will be the product ...

  area = (10 cm)²(1 +cos(θ))sin(θ)

__

This will be a maximum when its derivative is zero:

  d(area)/dθ = 0 = (100 cm²)(-sin(θ))sin(θ) +(1 +cos(θ))cos(θ)

  0 = cos(θ) -sin(θ)² +cos(θ)² . . . . . . simplify, divide by 100 cm²

  2cos(θ)² +cos(θ) -1 = 0 . . . . . . . use 1-cos² for sin²

  (2cos(θ)-1)(cos(θ) +1) = 0 . . . . . . factor

The solution of interest is ...

  cos(θ) = 1/2   ⇒   θ = π/3

The gutter will carry the maximum amount of water when the sides are bent up through an angles of π/3 radians.

_____

* The gutter is in the shape of a trapezoid. Its top (opening) dimension is (10 cm)(1+2cos(θ)), and its bottom width is 10 cm. The area of a trapezoid is half the sum of these values, multiplied by the height. Half the sum of the widths is ...

  ((10 +20cos(θ)) +10)/2 = 10 +10cos(θ) = 10(1 +cos(θ)) . . . . cm

You might be interested in
ebonee bought 16 chocolate bars for her 6 friends.If each friend receives an equal amount of chocolate bars. How many chocolate
Kamila [148]

Answer:

3/8

Step-by-step explanation:

6 0
3 years ago
Help me<br>ans should be well explained :) ​
Slav-nsk [51]
551, it is too big of a number so it is the outlier. :)
5 0
3 years ago
Read 2 more answers
PLZ HELPP 10 POINTS PLZZZ
kenny6666 [7]

Answer:

No,yes,yes,no,yes

Step-by-step explanation:

4 0
3 years ago
Factor 3x^4+5x^3-11x^2+3x
Andreas93 [3]

Answer:

x(x-1)(3x-1)(x+3)

Step-by-step explanation:

5 0
3 years ago
Please help i’ll give brainliest
Setler79 [48]

Answer:

range

Step-by-step explanation:

*JUST AND EDUCATED GUESS* but id say range

7 0
2 years ago
Other questions:
  • To make an international telephone call you need the code for the country you are calling. The codes for country A country B and
    10·1 answer
  • 4th question help please
    6·2 answers
  • Help please !!! show work​
    11·1 answer
  • Three baseball players are playing catch. Vondra is directly south of Mei and directly west of
    11·1 answer
  • Mangoes are on sale 4 for $3. Mindy buys 14 mangoes.
    9·1 answer
  • Find the general solution of (x+3)y’=2y
    10·1 answer
  • Helllllllllllllllllllp​
    6·1 answer
  • Multiply by 3 and add 1 help me plz :'(
    5·2 answers
  • (Pleaseeeeeee helpp me)
    12·2 answers
  • Solve for N. N - 6 = -7 N =
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!