Question

A landscaper is designing a flower garden in the shape of a trapezoid. She wants the shorter base to be 3 yards greater than the height, and the longer base to be 7 yards greater than the height. She wants the area to be 225 square yards. The situation is modeled by the equation. Use the Quadratic Formula to find the height that will give the desired area. Round to the nearest hundredth of a yard.

Answer 1

14.86 yd

Explanation:

• Put
  `h^(2)+5h=295` into standard form by subtracting 295 from both sides to become
  `h^(2) + 5h - 295 = 0`.
• Find the values of a, b, and c.
• a = 1, b = 5, and c = -295,
• Substitute those values into the quadratic formula
  `x = \frac{-b+/-\sqrt{b^(2)-4ac} }{2a}` and simplify.
• 
  `x=\frac{-5+/-\sqrt{(-5)^(2)- 4(1)(-295)} }{2(1)}`
• 
  `x=(-5+√(1205) )/(2) or x=(-5-√(1205) )/(2)`
• The second equation would give a negative solution, so only use the first equation and solve.
• 
  `x=(-5+√(1205) )/(2) = 14.86`

The height is 14.86 yd.

Answer 2

The formula for the area of a trapezium is as follows:

area=12(a+b)h

Let a be the shorter base. Then a = h + 3.
Let b be the longer base. Then b = h + 7.

Substituting these values for a and b in the general formula gives:
area = 225 = 12(h+3+h+7)h = h^2+5hSo you need to solve the following quadratic: h^2+5h−225=0
Step 1: Use quadratic formula with a=1, b=5, c=-225.

h=−b±√b2−4ac2ah=−(5)±√(5)2−4(1)(−225)2(1)h=−5±√9252h=−5/2+5/2√37 or h=−5/2+−5/2√37

Answer:h=−5/2+52√37 or h=−5/2+−5/2√37