5. Answer: rate of 6 mph + current of 5 mph = 11 mph
Step-by-step explanation:
Use d = r * t
[tex]\begin {array}{l|c|c|c||l} &\underline{time}&\underline{rate}&\underline{distance}&\underline{equation}\\ with&x&r+5&33&x(r+5)=33\\against&6-x&r-5&3&(6-x)(r-5)=3\\\end{array} \\\\\\\text{Solve each equation for x.}\\\\x(r + 5) = 33\qquad \qquad \qquad (6-x)(r-5)=3\\\\.\qquad \quad x=\dfrac{33}{r+5}\qquad \qquad \qquad \qquad 6-x=\dfrac{3}{r-5}\\\\\\.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x=-\dfrac{3}{r-5}+6[/tex]
[tex]\text{Now, set the equations equal to each other and solve for r:}\\\\\dfrac{33}{r+5}=-\dfrac{3}{r-5}+6\\\\\\\dfrac{33}{r+5}=-\dfrac{3}{r-5}+\dfrac{6(r-5)}{(r-5)}\\\\\\\dfrac{33}{r+5}=\dfrac{6r-33}{r-5}\\\\\\33(r-5)=(r+5)(6r-33)\\\\\\33r - 165 = 6r^2-3r-165\\\\\\.\qquad \quad 0=6r^2-36r\\\\\\.\qquad \quad 0=6r(r-6)\\\\\\.\qquad \quad 0=6r\quad or\quad 0=r-6\\\\\\.\qquad \quad r=0\quad or\quad r=6\\\\\text{0 mph is an erroneous solution so Karl's rate was 6 mph}[/tex]
6. Answer: 5 hours
Step-by-step explanation:
Use d = r * t. Since they both traveled the same distance, we can set the (rate * time) for each person equal to each other:
Wilbur: rate = 33 and time = x --> equation: 33x
Mary: rate = 55 and time = x - 2 --> equation: 55(x - 2)
33x = 55(x - 2)
33x = 55x - 110
0 = 22x - 110
110 = 22x
5 = x
