Factoring - Word Problem

Question

The area of a triangle is 77 inches^2.  If the height is 3 inches greater than the base, find the length of the base and the height. Note that the area of a triangle is A=1/2bh. Please show all steps.

Answer 1

Alisha,

 

 So, we have the formula for the area of a triangle: A=1/2bh, where b=base and h=height of triangle. Now, it's important to note that area here is in square inches, since the base and height are both in inches and inches*inches=inches². Saying that, I'll now suppress the units and work solely with the numbers. Since we know that the area is 77 and the height (h)  is (=)  3 more (+3)  inches than the base (b)      means    h=b+3  ,  we can now fill in what we know into the area formula to obtain:

 

 77=(1/2)(b)(b+3).  Now, simplifying gives us

 

77=(1/2)b²+(3/2)b. To get rid of denominators, we can multiply both sides by 2 (that's allowed since math tells us we can fiddle around with equations as long as we do the same thing to both sides!). So, doing that gives us

 

154=b²+3b. Since this is a quadratic equation and we're trying to solve for the base b, we need to move everything to one side. This gives us

 

b²+3b-154=0  , which factors into (b+14)(b-11)=0  , which gives us b=-14 and b=11. Since the base of a triangle can't have a negative measurement, b=-14 is an extraneous solution and is therefore disregarded. So, the base of our rectangle is 11 inches. Now, to get the height, we know that h=b+3. plugging in 11 for b gives us

 

h=11+3=14 inches. So, our base is 11 inches and our height is 14 inches. But, is this correct? We must check!

 

Does 77=(1/2)(11)(14) ?

 

Multiplying it out gives us 77=77, so yes that's indeed the correct answer.

 

Hope this helps!