Sally spent 6 hours walking from her house to the lake and back. She walked to the lake at a rate of 4mph and walked back at a rate of 2mph. What is the distance for her house to the lake?
Speed = Distance / time
v = s / t
In this case s = the distance for her house to the lake.
v1 = s / t1
She walked to the lake at a rate of 4 mph so:
v1 = s / t1
4 = s / t1
Multiply both sides by t1
4 t1 = s
Divide both sides by 4
t1 = s / 4
She walked back at a rate of 2mph so:
v2 = s / t2
2 = s / t2
Multiply both sides by t2
2 t2 = s
Divide both sides by 2
t2 = s / 2
Sally spent 6 hours walking from her house to the lake and back mean:
t1 + t2 = 6
s / 4 + s / 2 = 6
s / 4 + 2 s / 4 = 6
3 s / 4 = 6
Multiply both sides by 4
3 s = 24
Divide both sides by 3
s = 8
The distance for her house to the lake = 8 miles
Proof:
v1 = s / t1
t1 = s / v1
t1 = 8 / 4
t1 = 2 h
v2 = s / t2
t2 = s / v2
t2 = 8 / 2
t2 = 4 h
t1 + t2 = 2 h + 4 h = 6 h
Oh, Sally must have been determined to work on her cardio! Well, to calculate the distance from her house to the lake, we can use the formula: distance = speed × time.
Let's find out the time it took her to walk to the lake first. Given that she walked at a rate of 4mph for 6 hours, the distance she covered while going to the lake would be 4 × 6 = 24 miles.
Now, to calculate how long it took her to walk back from the lake, we use the same formula. This time, she walked at a rate of 2mph. Thus, the distance she covered while coming back from the lake would be 2 × 6 = 12 miles.
Since she walked the same distance both ways, we can conclude that the distance from her house to the lake is 24 miles.
To find the distance from Sally's house to the lake, we need to determine the total time it took for her to walk to and from the lake.
Let's assume that the distance between Sally's house and the lake is "D" miles.
The time it took for Sally to walk to the lake at a rate of 4mph can be calculated using the formula: Time = Distance / Rate
So, the time it took for Sally to walk to the lake is: Time1 = D miles / 4 mph = D/4 hours.
Similarly, the time it took for Sally to walk back from the lake at a rate of 2mph is: Time2 = D miles / 2 mph = D/2 hours.
The total time it took for Sally to walk from her house to the lake and back is given as 6 hours.
So, we can write the equation: Time1 + Time2 = 6 hours.
Plugging in the values, we have: D/4 + D/2 = 6.
To solve this equation, we can combine the like terms by finding a common denominator, which is 4, and then simplify the equation.
Multiplying the equation by its common denominator, we get: (D/4) * 4 + (D/2) * 4 = 6 * 4.
This simplifies to: D + 2D = 24.
Combining the like terms, we have: 3D = 24.
Dividing both sides of the equation by 3 gives us: D = 8.
Therefore, the distance from Sally's house to the lake is 8 miles.
To find the distance from Sally's house to the lake, we can use the formula: Distance = Speed x Time.
Let's denote the distance from her house to the lake as "d."
When Sally walks to the lake at a rate of 4 mph, the time it takes can be calculated by dividing the distance by the speed: Time to walk to the lake = d / 4.
Similarly, when she walks back from the lake at a rate of 2 mph, the time it takes can be calculated by dividing the distance by the speed: Time to walk back = d / 2.
We know that Sally spent a total of 6 hours walking to the lake and back. Therefore, we can write the equation: Time to walk to the lake + Time to walk back = 6.
Using the equations we derived earlier, we can substitute the expressions for the times: d / 4 + d / 2 = 6.
To find the value of d, we need to simplify and solve the equation.
First, let's add the terms on the left side: (d / 4) + (d / 2) = 6.
To add the fractions, we need a common denominator, which is 4 in this case. Rewriting the equation with the common denominator, we get: (d + 2d) / 4 = 6.
Combining like terms on the left side, we have: (3d) / 4 = 6.
To isolate d, we can multiply both sides of the equation by 4, which cancels out the denominator on the left side: 3d = 24.
Finally, we divide both sides by 3 to solve for d: d = 24 / 3.
Simplifying the expression, we find that d = 8.
Therefore, the distance from Sally's house to the lake is 8 miles.