SOLUTION: Have 2 questions. Question 1) The width of a rectangular gate is 2 meters (m) larger than its height. The diagonal brace is √6 m. Find the width and the height. Questi

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: Have 2 questions. Question 1) The width of a rectangular gate is 2 meters (m) larger than its height. The diagonal brace is √6 m. Find the width and the height. Questi      Log On

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Question 237201: Have 2 questions.
Question 1) The width of a rectangular gate is 2 meters (m) larger than its height. The diagonal brace is √6 m. Find the width and the height.
Question 2) The members of a flying club plan to share equally the cost of a $200,000 airplane. The members want to find five more people to join the club so that the cost per person will decrease by $2000. How many members are currently in the club?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Question 1) The width of a rectangular gate is 2 meters (m) larger than its height.
The diagonal brace is √6 m. Find the width and the height.
:
Let x = the height
then
(x+2) = the width
;
using pythag a^2 + b^2 = c^2
;
x^2 + (x+2)^2 = %28sqrt%286%29%29%5E2
FOIL
x^2 + x^2 + 4x + 4 = 6
:
2x^2 + 4x + 4 - 6 = 0
:
2x^2 + 4x - 2 = 0
Simplify divide by 2
x^2 + 2x - 1 = 0
Use the quadratic formula to find the positive solution:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
In this problem a=1; b=2; c=-1
x+=+%28-2+%2B-+sqrt%28+2%5E2-4%2A1%2A-1+%29%29%2F%282%2A1%29+
:
x+=+%28-2+%2B-+sqrt%284-+%28-4%29+%29%29%2F%282%29+
;
x+=+%28-2+%2B-+sqrt%284+%2B+4+%29%29%2F%282%29+
:
x+=+%28-2+%2B-+sqrt%288+%29%29%2F%282%29+
:
x+=+%28-2+%2B+2.828%29%2F%282%29+
:
x = .828%2F2
x = .414 meters is the height, 2.414 meters is the width
:
Check solution on a calc: enter sqrt%28.414%5E2+%2B+2.414%5E2%29 = 2.45 ~ sqrt%286%29
;
;
Question 2) The members of a flying club plan to share equally the cost of a $200,000 airplane.
The members want to find five more people to join the club so that the cost per person will decrease by $2000.
How many members are currently in the club?
:
Let x = no. of members in the club now
then
(x+5) = no. of members to decrease cost per person by 2k
:
To reduce typing lets do this in "thousands of dollars"
;
Current cost per member: 200%2Fx
:
current cost - more member cost = 2 thousand
200%2Fx - 200%2F%28%28x%2B5%29%29 = 2
Multiply equation by x(x+5); results:
200(x+5) - 200x = 2x(x+5)
:
200x + 1000 - 200x = 2x^2 + 10x
:
1000 = 2x^2 + 10x
:
A quadratic equation
2x^2 + 10x - 1000 = 0
:
Simplify, divide by 2
x^2 + 5x - 500 = 0
;
Factors easily to
(x+25)(x-20) = 0
:
Positive solution
x = 20 members currently
:
Check solution:
200/20 = 10 thousand per member
200/25 = 8 thousand per member
--------------------
Saving: 2k per member
: