Simplify the sum. State any restrictions on the variables. (3 points)
x-2/x+3 + 10x/x²-9
The speed of the current in a river is 6 mph. A ferry operator who works that part of the river is looking to buy a new boat for his business. Every day, his route takes him 22.5 miles each way against the current and back to his dock, and he needs to make this trip in a total of 9 hours. He has a boat in mind, but he can only test it on a lake where there is no current. How fast must the boat go on the lake in order for it to serve the ferry operator's needs?
if you don't want to give answer can you use a similar problem and work it out. I wouldn't mind understanding what I'm doing instead of going to the internet but my teacher teaches with PowerPoints... I can't learn math from a power point I need step by step and pencil and paper. With this in mind, your help will be greatly appreciated
Ferry boat must go 9mph!!
Sure, I understand that learning math through PowerPoints can be challenging. Let's work through a similar problem step by step.
Let's assume that the speed of the boat on the lake is "b" mph. Since there is no current in the lake, the boat's speed remains constant.
When the ferry is moving against the current, the speed of the boat relative to the water is the difference between the boat's speed and the current's speed. So, the speed of the boat against the current is "b - 6" mph.
On his route, the ferry needs to travel 22.5 miles against the current, and 22.5 miles with the current. The total distance covered in the round trip is 45 miles.
The time taken to travel a certain distance is equal to the distance divided by the speed. So, the time taken to travel 22.5 miles against the current is 22.5 / (b - 6), and the time taken to travel 22.5 miles with the current is 22.5 / (b + 6).
According to the problem, the total time taken for the round trip is 9 hours. So, we can set up the equation:
22.5 / (b - 6) + 22.5 / (b + 6) = 9
Now, let's simplify and solve the equation:
To eliminate the fractions, let's multiply both sides of the equation by (b - 6)(b + 6):
22.5(b + 6) + 22.5(b - 6) = 9(b - 6)(b + 6)
Expanding the brackets:
22.5b + 135 + 22.5b - 135 = 9(b^2 - 36)
Combining like terms:
45b = 9b^2 - 324
Rearranging the equation to set it equal to zero:
9b^2 - 45b - 324 = 0
Dividing both sides by 9:
b^2 - 5b - 36 = 0
Now, we can factor the quadratic equation:
(b - 9)(b + 4) = 0
Setting each factor equal to zero:
b - 9 = 0 or b + 4 = 0
Solving for "b" in each case:
b = 9 or b = -4
Since speed cannot be negative, we can discard the solution b = -4.
Therefore, the boat must go at a speed of 9 mph on the lake in order to serve the ferry operator's needs.
I hope this step-by-step explanation helps! Let me know if you have any further questions.
To simplify the given sum, we'll need to find the common denominator and combine the fractions.
The first fraction is x - 2 / x + 3. The denominator is x + 3.
The second fraction is 10x / (x² - 9). The denominator is x² - 9, which is a difference of squares and can be factored as (x + 3)(x - 3).
To find the common denominator, we need to multiply the denominators of both fractions, so we have (x + 3)(x - 3)(x + 3) = (x + 3)²(x - 3).
Now, we can rewrite the two fractions with the common denominator:
(x - 2)(x - 3) / (x + 3)(x - 3) + 10x / (x + 3)²(x - 3)
Next, we can combine the fractions by adding the numerators since they have the same denominator:
[(x - 2)(x - 3) + 10x] / (x + 3)²(x - 3)
To simplify further, distribute the (x - 2) term:
[(x² - 5x + 6) + 10x] / (x + 3)²(x - 3)
Now combine like terms in the numerator:
[x² + 5x + 6] / (x + 3)²(x - 3)
Thus, the simplified sum is (x² + 5x + 6) / (x + 3)²(x - 3). No restrictions on the variable were mentioned in the original question, so there are no restrictions on x.
Now, let's move on to the second problem about the ferry operator.
Similar Problem:
A ferry operator needs to make a round trip of 60 miles along a river with a speed of 8 mph against the current. The current speed is 4 mph. The operator can test the boat on a lake with no current. What speed must the boat go on the lake to complete the round trip in a total of 10 hours?
Solution:
To solve this problem, we'll use the formula: distance = speed × time.
Let's consider the time taken to go against the current. The distance is 30 miles (half of the round trip) and the speed is (boat speed - current speed) = (boat speed - 4 mph). Let's denote this speed as "s":
30 = (s - 4) × time against the current [Equation 1]
The time against the current is calculated using the formula: time = distance / speed.
Substituting the values into the equation:
time against the current = 30 / (s - 4)
Now, let's consider the time taken to go with the current. The distance is also 30 miles (the second half of the round trip) and the speed is (boat speed + current speed) = (boat speed + 4 mph). Let's denote this speed as "w":
30 = (w + 4) × time with the current [Equation 2]
The time with the current is calculated using the formula: time = distance / speed.
Substituting the values into the equation:
time with the current = 30 / (w + 4)
The total time for the round trip is given as 10 hours:
time against the current + time with the current = 10
Substituting the values of time against the current and time with the current into the equation:
30 / (s - 4) + 30 / (w + 4) = 10
Now, you can solve this equation to find the value of "s" (the speed of the boat on the lake) and "w" (the speed of the boat with the current). This will allow you to answer the original question.
To simplify the sum, we first need to find a common denominator for the two fractions. The denominator of the first fraction is (x+3), and the denominator of the second fraction is (x²-9).
Since (x²-9) can be factored as (x-3)(x+3), the common denominator is (x+3)(x-3).
Now, let's rewrite the fractions with the common denominator:
(x-2)/(x+3) + 10x/(x²-9) = [(x-2)(x-3)]/[(x+3)(x-3)] + 10x/[(x+3)(x-3)]
Next, we can combine the fractions over the common denominator:
[(x-2)(x-3) + 10x]/[(x+3)(x-3)]
Simplifying the numerator:
(x² - 5x + 6 + 10x)/[(x+3)(x-3)] = (x² + 5x + 6)/[(x+3)(x-3)]
So, the simplified sum is (x² + 5x + 6)/[(x+3)(x-3)].
As for restrictions on the variables, we need to consider the denominator. The denominators (x+3) and (x-3) cannot be equal to zero. Therefore, we have the following restrictions:
x+3 ≠ 0
x ≠ -3
x-3 ≠ 0
x ≠ 3
So, the restrictions on the variables are x ≠ -3 and x ≠ 3.
Now, let's move on to the second part of your question regarding the boat's speed.
To determine how fast the boat must travel on the lake, we need to consider the total distance traveled by the ferry operator, the speed of the current, and the time taken.
The distance traveled against the current is 22.5 miles, and the time taken is 9 hours for a round trip. Since the speed of the current is 6 mph, we need to account for the fact that the boat will be moving slower when going against the current and faster when moving with the current.
Let's assume the speed of the boat on the lake is b mph. When traveling against the current, the effective speed of the boat will be (b - 6) mph, and when traveling with the current, the effective speed will be (b + 6) mph.
So, for the 22.5-mile journey against the current, the time taken can be calculated using the formula:
time = distance / speed
Substituting the values:
22.5 = 22.5 / (b - 6)
Simplifying:
b - 6 = 1
b = 7
Therefore, the ferry operator's boat must have a speed of 7 mph on the lake in order to make the 22.5-mile trip against the current in 9 hours.
I hope this step-by-step explanation helps you understand the process better. If you have any further questions, feel free to ask!
You need brackets for this question to work out
(x-2)/(x+3) + 10x/(x²-9)
= (x-2)/(x+3) + 10x/((x-3)(x+3)) , so the LCD is (x+3)(x-3)
= ( (x-2)(x-3) + 10x) / ((x-3)(x+3))
= (x^2 - 5x + 6 + 10x) / ((x-3)(x+3))
= (x^2 + 5x + 6) / ((x-3)(x+3)) , x ≠ ±3
Your river question:
the unknown is the speed of the boat he wants to buy
let the speed of the boat be x mph
So the effective speed of the boat against the current will be x - 6 mph
the effective speed of the boat with the current will be x + 6 mph
time going with the current = 22.5/(x+6)
time going against the current = 22.5/(x-6)
we want:
22.5/(x+6) + 22.5/(x-6) = 9
multiply each term by (x=6)(x-6) , the LCD of our two fractions
22.5(x-6) + 22.5(x+6) = 9(x+6)(x-6)
22.5x - 135 + 22.5x + 135 = 6x^2 - 324
6x^2 - 45x - 324 = 0
2x^2 - 15x - 108 = 0
(x - 12)(2x + 9) = 0
so x = 12, or x = -9/2, which of course makes no sense
He should buy a boat that can do at least 12 mph