Question

1. on monday, you run on a treadmill for 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 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:

9.

Step1: Find distance - Monday

Use the formula $d = vt$ (distance = velocity×time). On Monday, the time $t_1=\frac{1}{2}$ hour and the speed $v_1 = x$ miles per hour. So the distance $d_1=\frac{1}{2}x$ miles.

Step2: Find distance - Tuesday

On Tuesday, the speed $v_2=(x - 2)$ miles per hour and the time $t_2=\frac{3}{4}$ hour. So the distance $d_2=\frac{3}{4}(x - 2)$ miles.

Step3: Set up equation

Since $d_1 = d_2$, we have the equation $\frac{1}{2}x=\frac{3}{4}(x - 2)$.
Multiply both sides 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 $x = 6$.

Step4: Calculate Monday's distance

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

Step1: Set up equation

Since Jess and her sister spent the same amount of time on the computer, we set up the equation $7x=5x + 10$.

Step2: Solve for x

Subtract $5x$ from both sides: $7x-5x=5x + 10-5x$.
We get $2x=10$.
Divide both sides by 2: $x = 5$.

Step3: Find Jess's time

Substitute $x = 5$ into the expression for Jess's time, which is $7x$. So $7\times5=35$ minutes.

Step1: Find rectangle's area formula

The area of a rectangle $A_{r}=l\times w$, where $w = 6$ and $l=(x - 8)$. So $A_{r}=6(x - 8)=6x-48$.

Step2: Find triangle's area formula

The area of a triangle $A_{t}=\frac{1}{2}\times b\times h$, where $h = 7$ and $b=(x - 3)$. So $A_{t}=\frac{1}{2}\times7\times(x - 3)=\frac{7}{2}(x - 3)=\frac{7x-21}{2}$.

Step3: Set up equation

Since $A_{r}=A_{t}$, we have $6x-48=\frac{7x - 21}{2}$.
Multiply both sides by 2: $2(6x-48)=7x - 21$.
Expand the left - hand side: $12x-96=7x - 21$.
Subtract $7x$ from both sides: $12x-7x-96=7x-7x - 21$.
We get $5x-96=-21$.
Add 96 to both sides: $5x-96 + 96=-21 + 96$.
So $5x=75$, and $x = 15$.

Step4: Calculate rectangle's area

Substitute $x = 15$ into the rectangle's area formula $A_{r}=6(x - 8)$.
$A_{r}=6\times(15 - 8)=6\times7 = 42$ square units.

Answer:

3 miles

10.