# 1. In order to collect data on the signal strengths in a neighborhood, Briana must drive from house to house and take readings. She has a graduate student, Henry, to assist her in the task. Briana figures it would take her 12 hours to complete the task if working alone, and that it would take Henry 18 hours if he were to complete the task by himself.

How long will it take Briana and Henry if they complete the task together?

2. Briana can collect data from 5 more houses per hour than Henry. It took Briana two hours less to collect data from 120 houses than it took Henry to collect data from the same number of houses.
3. Joe is the owner and CEO of Pinecrest Enterprises, a real estate development company in Boise, Idaho. The company has taken on a new development project outside of town, and Joe wants to drive out to survey the proposed development site. The site is 30 miles away from Joe’s office. Setting a good example, Joe drives the speed limit of 45 miles per hour to the development site and back.
4. The development site consists of a rectangular plot of land, with a smaller rectangle set aside as National Forest land (which is not part of the site). Driving north along the eastern edge of the site, Joe uses the odometer in his truck to determine the length of the forest is 1.2 miles, and the remaining length of the site is 2.0 miles. He also knows the width of the forest is one mile less than the width of the southern portion of the site. (See the figure below.) The total area of the site excluding the forest is 16 square miles.
What is the width, x, of region 2, as shown in the picture above?

## #1

1/12 + 1/18 = 1/x

Now, what do you think about the other parts? You must have some ideas about at least some of them.

## 1. To calculate how long it will take Briana and Henry to complete the task together, we can use the formula for their combined work rate. Let's call their combined work rate R.

Briana's work rate: 1/12 of the task per hour (since she can complete the task in 12 hours)
Henry's work rate: 1/18 of the task per hour (since he can complete the task in 18 hours)

Their combined work rate: Briana's work rate + Henry's work rate = 1/12 + 1/18

To find the time it will take both of them to complete the task together, we can take the reciprocal of their combined work rate:

Time = 1 / (1/12 + 1/18)

Simplifying this expression:

Time = 1 / (3/36 + 2/36)
= 1 / (5/36)
= 36/5
= 7.2 hours

Therefore, Briana and Henry can complete the task together in 7.2 hours.

2. Let's assume that Briana's data collection rate is B houses per hour, and Henry's data collection rate is H houses per hour.

According to the given information, Briana can collect data from 5 more houses per hour than Henry, so we can write the equation:

B = H + 5

It took Briana two hours less to collect data from 120 houses than it took Henry, so we can write the equation:

2 = 120/B - 120/H

Now, we can substitute the value of B from the first equation into the second equation:

2 = 120/(H + 5) - 120/H

To simplify this equation, we can find a common denominator:

2 = (120H - 120(H + 5))/(H(H + 5))

Simplifying further:

2 = (120H - 120H - 600)/(H^2 + 5H)
2 = -600/(H^2 + 5H)

Multiplying both sides of the equation by (H^2 + 5H):

2H^2 + 10H = -600

Rearranging the equation:

2H^2 + 10H + 600 = 0

This is a quadratic equation. To solve for H, we can use the quadratic formula:

H = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 2, b = 10, and c = 600. Plugging in these values and solving for H:

H = (-10 ± √(10^2 - 4 * 2 * 600))/(2 * 2)
H = (-10 ± √(100 - 4800))/4
H = (-10 ± √(-4700))/4

Since the discriminant is negative, there are no real solutions for H. This means that there is no possible value for H that satisfies the given conditions.

## 1. To find out how long it will take Briana and Henry to complete the task together, we can use the formula for their combined work rate. The formula is given by:

1/Briana's work rate + 1/Henry's work rate = 1/Combined work rate

From the given information, we know that it would take Briana 12 hours to complete the task by herself, so her work rate can be calculated as 1/12. Similarly, Henry's work rate is 1/18.

Substituting these values into the formula, we get:

1/12 + 1/18 = 1/Combined work rate

To simplify this equation, we need to find a common denominator for 12 and 18, which is 36. Therefore:

(3/36) + (2/36) = 1/Combined work rate

Combining the fractions, we have:

5/36 = 1/Combined work rate

To isolate the combined work rate, we take the reciprocal of both sides of the equation:

Combined work rate = 36/5

Finally, to find out how long it will take Briana and Henry to complete the task together, we take the reciprocal of the combined work rate:

Combined time = 5/36 hours

2. Let's solve this problem by setting up equations based on the given information.

Let's say Briana's data collection rate is "x" houses per hour. Therefore, Henry's data collection rate would be "x - 5" houses per hour.

It took Briana two hours less than Henry to collect data from 120 houses. This can be represented as:

120 / (x + 5) - 2 = 120 / x

To solve for x, we can cross multiply:

120x = (120 / (x + 5) - 2) * x

Further simplifying the equation:

120x = (120 - 2x - 10) * x

120x = 120x - 2x^2 - 10x

Rearranging the equation:

2x^2 + 10x - 120x = 0

Simplifying:

2x^2 - 110x = 0

Factoring out 2x:

2x (x - 55) = 0

Setting each factor equal to zero:

2x = 0 or x - 55 = 0

From the first equation, we get:

x = 0

However, it doesn't make sense for x to be zero, so we can ignore this solution.

From the second equation, we get:

x - 55 = 0

Solving for x, we find:

x = 55

Therefore, Briana's data collection rate is 55 houses per hour, and Henry's data collection rate is 55 - 5 = 50 houses per hour.

3. To determine the time it takes Joe to drive to the development site and back, you need to calculate the total distance he travels. Joe drives at a speed of 45 miles per hour, and the site is 30 miles away.

The distance from Joe's office to the development site is 30 miles. Since he has to drive there and back, the round trip distance can be calculated as:

Round trip distance = 2 * Distance from office to site = 2 * 30 = 60 miles.

To find the time it takes Joe to travel this distance, we can use the formula:

Time = Distance / Speed

Plugging in the values, we get:

Time = 60 miles / 45 miles per hour = 1.33 hours (rounded to two decimal places).

So, it takes Joe approximately 1.33 hours to drive to the development site and back at the speed limit.

4. To find the width, x, of region 2, we need to gather information from the given figure and problem statement.

We are given that the total area of the site, excluding the forest, is 16 square miles. This is represented by the shaded region in the figure.

The total area of a rectangle can be calculated by multiplying its length and width. In this case, the length is the given length of the site, which is 2.0 miles.

Let's represent the unknown width of the southern portion of the site as y.

The area of the site excluding the forest can be expressed as (2.0 - 1.2)(y) = 16.

Simplifying the equation, we have:

0.8y = 16

To find y, we divide both sides of the equation by 0.8:

y = 16 / 0.8

y = 20

Therefore, the width of the southern portion of the site is 20 miles.

From the figure, it is given that the width of the forest is one mile less than the width of the southern portion of the site. So, the width of the forest is 20 - 1 = 19 miles.

The width of region 2 can be found by subtracting the width of the forest from the given length of the site:

x = 2.0 - 19

x = -17 (since 19 is greater than 2)

However, a negative width does not make sense in this context, so there may be an error in the given information or figure. Please double-check the problem statement or figure to ensure the accuracy of the dimensions provided.