Answer:
The length of the base is 11 meters.
Step-by-step explanation:
The diagram of the triangle is not shown; However, the given details are enough to solve this question.
Given
<em>Shape: Triangle</em>
<em>Represent the height with h and the base with b</em>
![b = 3 + 2h](https://tex.z-dn.net/?f=b%20%3D%203%20%2B%202h)
![Area = 22](https://tex.z-dn.net/?f=Area%20%3D%2022)
Required
Find the length of the base
The area of a triangle is calculated as thus;
![Area = \frac{1}{2} * b * h](https://tex.z-dn.net/?f=Area%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20b%20%2A%20h)
Substitute 22 for Area and 3 + 2h for b
The formula becomes
![22 = \frac{1}{2} * (3 + 2h) * h](https://tex.z-dn.net/?f=22%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20%283%20%2B%202h%29%20%2A%20h)
Multiply both sides by 2
![2 * 22 = 2 * \frac{1}{2} * (3 + 2h) * h](https://tex.z-dn.net/?f=2%20%2A%2022%20%3D%202%20%2A%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20%283%20%2B%202h%29%20%2A%20h)
![44 = (3 + 2h) * h](https://tex.z-dn.net/?f=44%20%3D%20%283%20%2B%202h%29%20%2A%20h)
Open the bracket
![44 = 3 * h + 2h * h](https://tex.z-dn.net/?f=44%20%3D%203%20%2A%20h%20%2B%202h%20%2A%20h)
![44 = 3h + 2h^2](https://tex.z-dn.net/?f=44%20%3D%203h%20%2B%202h%5E2)
Subtract 44 from both sides
![44 - 44 = 3h + 2h^2 - 44](https://tex.z-dn.net/?f=44%20-%2044%20%3D%203h%20%2B%202h%5E2%20-%2044)
![0 = 3h + 2h^2 - 44](https://tex.z-dn.net/?f=0%20%3D%203h%20%2B%202h%5E2%20-%2044)
Rearrange
![0 = 2h^2 +3h - 44](https://tex.z-dn.net/?f=0%20%3D%202h%5E2%20%2B3h%20-%2044)
![2h^2 +3h - 44 = 0](https://tex.z-dn.net/?f=2h%5E2%20%2B3h%20-%2044%20%3D%200)
At this point, we have a quadratic equation; which is solved as follows:
![2h^2 +3h - 44 = 0](https://tex.z-dn.net/?f=2h%5E2%20%2B3h%20-%2044%20%3D%200)
![2h^2 + 11h - 8h - 44 = 0](https://tex.z-dn.net/?f=2h%5E2%20%2B%2011h%20-%208h%20-%2044%20%3D%200)
![h(2h + 11) - 4(2h + 11) = 0](https://tex.z-dn.net/?f=h%282h%20%2B%2011%29%20-%204%282h%20%2B%2011%29%20%3D%200)
![(h - 4)(2h + 11) = 0](https://tex.z-dn.net/?f=%28h%20-%204%29%282h%20%2B%2011%29%20%3D%200)
Split the above
![(h - 4) = 0\ or\ (2h + 11) = 0](https://tex.z-dn.net/?f=%28h%20-%204%29%20%3D%200%5C%20or%5C%20%282h%20%2B%2011%29%20%3D%200)
![h - 4 = 0\ or\ 2h + 11 = 0](https://tex.z-dn.net/?f=h%20-%204%20%3D%200%5C%20or%5C%202h%20%2B%2011%20%3D%200)
Solve the above linear equations separately
![h - 4 = 0](https://tex.z-dn.net/?f=h%20-%204%20%3D%200)
Add 4 to both sides
![h - 4 + 4 = 0 + 4](https://tex.z-dn.net/?f=h%20-%204%20%2B%204%20%3D%200%20%2B%204)
![h = 0 + 4](https://tex.z-dn.net/?f=h%20%3D%200%20%2B%204)
---- <em>First value of h</em>
![2h + 11 = 0](https://tex.z-dn.net/?f=2h%20%2B%2011%20%3D%200)
Subtract 11 from both sides
![2h + 11 - 11 = 0 - 11](https://tex.z-dn.net/?f=2h%20%2B%2011%20-%2011%20%3D%200%20-%2011)
![2h = 0 - 11](https://tex.z-dn.net/?f=2h%20%20%3D%200%20-%2011)
![2h = -11](https://tex.z-dn.net/?f=2h%20%3D%20-11)
Divide both sides by 2
![\frac{2h}{2} = -\frac{11}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B2h%7D%7B2%7D%20%3D%20-%5Cfrac%7B11%7D%7B2%7D)
<em> ------ Second value of h</em>
Since height can be negative, we'll discard ![h = -\frac{11}{2}](https://tex.z-dn.net/?f=h%20%3D%20-%5Cfrac%7B11%7D%7B2%7D)
Hence, the usable value of height is ![h = 4](https://tex.z-dn.net/?f=h%20%3D%204)
Recall that ![b = 3 + 2h](https://tex.z-dn.net/?f=b%20%3D%203%20%2B%202h)
Substitute 4 for h
![b = 3 + 2(4)](https://tex.z-dn.net/?f=b%20%3D%203%20%2B%202%284%29)
![b = 3 + 8](https://tex.z-dn.net/?f=b%20%3D%203%20%2B%208)
![b = 11](https://tex.z-dn.net/?f=b%20%3D%2011)
Hence, the length of the base is 11 meters