Question

1. on monday, you run on a treadmill for $\frac{1}{2}$ hour at $x$ miles per hour. on tuesday, you walk the same distance on the treadmill, at 2 miles per hour slower, and it takes you $\frac{3}{4}$ hour. how many miles did you run on the treadmill on monday? 10. jess spent 7x minutes on the computer. her sister spent (5x + 10) minutes on the computer, which was the same amount of time jess spent. how many minutes was jess on the computer? 11. a rectangle is 6 units wide and (x - 8) units long. it has the same area as a triangle with a height of 7 units and a base of (x - 3) units. what is the area of the rectangle?

Explanation:

Step1: Set up distance - rate - time equation for Monday and Tuesday

On Monday, distance $d_1=\frac{1}{2}x$ (using $d = rt$, where $r = x$ and $t=\frac{1}{2}$). On Tuesday, rate $r_2=x - 2$ and time $t_2=\frac{3}{4}$, so distance $d_2=\frac{3}{4}(x - 2)$. Since $d_1=d_2$, we have $\frac{1}{2}x=\frac{3}{4}(x - 2)$.

Step2: Solve the equation for $x$

Multiply both sides of the equation $\frac{1}{2}x=\frac{3}{4}(x - 2)$ by 4 to get $2x = 3(x - 2)$. Expand the right - hand side: $2x=3x - 6$. Subtract $2x$ from both sides: $0 = 3x-2x - 6$, so $x = 6$.

Step3: Find the distance run on Monday

Substitute $x = 6$ into the Monday distance formula $d_1=\frac{1}{2}x$. Then $d_1=\frac{1}{2}\times6 = 3$ miles.

Step4: Solve problem 10

Set up the equation $7x=5x + 10$. Subtract $5x$ from both sides: $7x-5x=5x + 10-5x$, $2x=10$, so $x = 5$. Jess spent $7x$ minutes on the computer, substituting $x = 5$, she spent $7\times5=35$ minutes.

Step5: Solve problem 11

Area of rectangle $A_{rect}=6(x - 8)$. Area of triangle $A_{tri}=\frac{1}{2}\times7\times(x - 3)$. Set $A_{rect}=A_{tri}$, so $6(x - 8)=\frac{7}{2}(x - 3)$. Multiply both sides by 2 to get $12(x - 8)=7(x - 3)$. Expand: $12x-96=7x - 21$. Subtract $7x$ from both sides: $12x-7x-96=7x-7x - 21$, $5x-96=-21$. Add 96 to both sides: $5x=-21 + 96$, $5x=75$, so $x = 15$. Area of rectangle $A_{rect}=6\times(15 - 8)=6\times7 = 42$ square units.

Answer:

1. 3 miles
2. 35 minutes
3. 42 square units