Respuesta :
Using the together rate, it is found that:
- It takes 3 hours for the first handyman to do the home repair individually.
- It takes 6 hours for the second handyman.
The together rate is the sum of each separate rate.
In this problem, we have that:
- The together rate is of [tex]\frac{1}{2}[/tex].
- The first man takes x hours, hence his rate is of [tex]\frac{1}{x}[/tex]
- The second man takes x + 3 hours, hence his rate is of [tex]\frac{1}{x + 3}[/tex]
Applying the together rate:
[tex]\frac{1}{x} + \frac{1}{x + 3} = \frac{1}{2}[/tex]
[tex]\frac{x + 3 + x}{x(x + 3)} = \frac{1}{2}[/tex]
[tex]4x + 6 = x^2 + 3x[/tex]
[tex]x^2 - x - 6 = 0[/tex]
Which is a quadratic equation with coefficients [tex]a = 1, b = -1, c = -6[/tex], hence:
[tex]\Delta = b^2 - 4ac = (-1)^2 - 4(1)(-6) = 25[/tex]
[tex]x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{1 + \sqrt{25}}{2} = 3[/tex]
[tex]x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{1 - \sqrt{25}}{2} = -2[/tex]
We are interested into the positive root, then [tex]x = 3[/tex], [tex]x + 3 = 6[/tex], which means that:
- It takes 3 hours for the first handyman to do the home repair individually.
- It takes 6 hours for the second handyman.
You can learn more about the together rate at https://brainly.com/question/25159431
