SOLUTION: Help! My final is on Monday and I can't figure out this word problem. Please write it out step by step so I will be able to do it on the final. Thank you. The height of a triangle

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Help! My final is on Monday and I can't figure out this word problem. Please write it out step by step so I will be able to do it on the final. Thank you. The height of a triangle      Log On


   

Question 173140: Help! My final is on Monday and I can't figure out this word problem. Please write it out step by step so I will be able to do it on the final. Thank you.
The height of a triangle is 8 centimeters greter than the base. The area of the triangle is 90 square centimeters. Find the length of the base and height of the triangle. Define a variable. Set up a quadratic equation. Solve. Show all work.
I'm really stuck, I thought I knew it but I've been at it most of the day.
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let b=base and h=height

Since "The height of a triangle is 8 centimeters greter than the base", this means that h=b%2B8

Also, remember that the formula for the area of a triangle is

A=%28b%2Ah%29%2F2

Since the "area of the triangle is 90 square centimeters", this tells us that A=90


90=%28b%28b%2B8%29%29%2F2 Plug in A=90 (the given area) and h=b%2B8


2%2A90=b%28b%2B8%29 Multiply both sides by 2


180=b%28b%2B8%29 Multiply


180=b%5E2%2B8b Distribute


0=b%5E2%2B8b-180 Subtract 180 from both sides.



Notice we have a quadratic equation in the form of ab%5E2%2Bbb%2Bc where a=1, b=8, and c=-180


Let's use the quadratic formula to solve for b


b+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


b+=+%28-%288%29+%2B-+sqrt%28+%288%29%5E2-4%281%29%28-180%29+%29%29%2F%282%281%29%29 Plug in a=1, b=8, and c=-180


b+=+%28-8+%2B-+sqrt%28+64-4%281%29%28-180%29+%29%29%2F%282%281%29%29 Square 8 to get 64.


b+=+%28-8+%2B-+sqrt%28+64--720+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-180%29 to get -720


b+=+%28-8+%2B-+sqrt%28+64%2B720+%29%29%2F%282%281%29%29 Rewrite sqrt%2864--720%29 as sqrt%2864%2B720%29


b+=+%28-8+%2B-+sqrt%28+784+%29%29%2F%282%281%29%29 Add 64 to 720 to get 784


b+=+%28-8+%2B-+sqrt%28+784+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


b+=+%28-8+%2B-+28%29%2F%282%29 Take the square root of 784 to get 28.


b+=+%28-8+%2B+28%29%2F%282%29 or b+=+%28-8+-+28%29%2F%282%29 Break up the expression.


b+=+%2820%29%2F%282%29 or b+=++%28-36%29%2F%282%29 Combine like terms.


b+=+10 or b+=+-18 Simplify.


So the possible lengths of the base are b+=+10 or b+=+-18


However, a negative base does NOT make sense. So this means that the only possible length for the base is b=10


So the base is 10 centimeters.


h=b%2B8 Go back to the height equation (remember the height is 8 cm longer than the base)


h=10%2B8 Plug in b=10


h=18 Add


So the height is 18 cm


===============================================================

Answer:

So the base is 10 cm and the height is 18 cm.




Check:


Area: A=%2810%2A18%29%2F2=180%2F2=90

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let base = b
height = b+%2B+8
90+=+%281%2F2%29%2Ab%2A%28b+%2B+8%29
multiply both sides buy 2
180+=+b%5E2+%2B+8b
Complete the square
b%5E2+%2B+8b+%2B+%288%2F2%29%5E2+=+180+%2B+%288%2F2%29%5E2
%28b+%2B+4%29%5E2+=+196
take square root of both sides
b+%2B+4+=+14
b+=+10
b+%2B+8+=+18
The base is 10 cm and the height is 18 cm
check:
90+=+%281%2F2%29%2Ab%2Ah
90+=+%281%2F2%29%2A10%2A18
90+=+180%2F2
90+=+90
OK