1. The speed of a 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?
MY ANSWER: NO IDEA
2. The sum of the reciprocals of two consecutive positive integers is 17/72. Write an equation that can be used to find the two integers. What are the integers?
MY ANSWER: NO IDEA
[(x^2-16)/(x^2+5x+6)]
_______________________
[(x^2+5x+4)/(x^2-2x-8)]
^^^^^its supposed to be a fraction
3. What is the quotient in simplified form? State any restrictions on the variable.
a. [(x-4)^2]/[(x+3)(x+1)]; x cannot equal -3 or -1
b. [(x-4)^2]/[(x+3)(x+1)]; x cannot equal -4,-3,-2,-1, or 4
MY ANSWER: I think it is A, but I thought that that restrictions were -4,-3,-2, and -1 but not 4.
since time = distance/speed,
22.5/(x-6) + 22.5/(x+6) = 9
1/x + 1/(x+1) = 17/72
hint: 17 = 8+9
To get rid of the fraction in the denominator, invert and multiply. Then you have
(x^2-16)/(x^2+5x+6) * (x^2-2x-8)/(x^2+5x+4)
=
(x+4)(x-4)(x+2)(x-4)
-------------------------------
(x+2)(x+3)(x+1)(x+4)
Hmm. You are correct. A is missing -2 and -4, but B incorrectly excludes x=4.
1. To calculate how fast the boat needs to go on the lake, we need to consider the time it takes for the boat to travel against the current and with the current. Let's denote the speed of the boat on the lake as "v" mph.
Against the Current:
The time taken to travel 22.5 miles against the current can be calculated using the formula: Time = Distance / Speed. So, the time taken against the current is 22.5 / (v - 6) hours.
With the Current:
The time taken to travel 22.5 miles with the current can be calculated using the same formula: Time = Distance / Speed. So, the time taken with the current is 22.5 / (v + 6) hours.
Total Time:
The total time taken to complete the round trip is given as 9 hours. So, the equation becomes:
22.5 / (v - 6) + 22.5 / (v + 6) = 9
Now we can solve this equation to find the value of "v" which represents the speed of the boat on the lake.
2. Let's assume the two consecutive positive integers as "x" and "x+1".
The sum of their reciprocals is given as 17/72, so we can write the equation as:
1/x + 1/(x+1) = 17/72
To solve this equation, we can find a common denominator and simplify. Multiplying through by 72x(x+1), we get:
72(x+1) + 72x = 17x(x+1)
Expanding and rearranging the equation, we have:
144x + 72 = 17x^2 + 17x
Now, let's bring all terms to one side of the equation and simplify:
17x^2 + 17x - 144x - 72 = 0
17x^2 - 127x - 72 = 0
You can solve this quadratic equation using factorization, completing the square, or using the quadratic formula to find the values of "x" and "x+1".
3. Let's simplify the given fraction:
a. [(x-4)^2] / [(x+3)(x+1)]
The numerator can be expanded:
[(x-4)(x-4)] = (x-4)^2 = x^2 - 8x + 16
The denominator can be multiplied:
[(x+3)(x+1)] = x^2 + 4x + 3
So, the fraction becomes:
(x^2 - 8x + 16) / (x^2 + 4x + 3)
Restrictions on the variable exist when the denominator becomes zero. Thus, we need to find the values of "x" that make the denominator equal to zero.
For part a: (x^2 + 4x + 3) ≠ 0
(x+3)(x+1) ≠ 0
This condition holds true for all values of "x" except x = -3 and x = -1. Therefore, the restrictions on the variable are x ≠ -3 and x ≠ -1.
For part b, the restrictions on the variable are x ≠ -4, -3, -2, -1, and 4.
Please double-check the answer to question 3 according to the given restrictions.
1. To determine the required speed of the boat on the lake, we need to consider the time it takes for the ferry operator to travel against the current and the time it takes to travel with the current.
Let's break down the information given:
- The round trip is 22.5 miles each way, so the total distance traveled is 45 miles.
- The speed of the current is 6 mph.
- The ferry operator needs to make the round trip in a total of 9 hours.
When the boat is traveling against the current, its effective speed is reduced by the speed of the current. When the boat is traveling with the current, its effective speed is increased by the speed of the current.
Now, let's set up the equation to find the speed of the boat on the lake:
Let b be the speed of the boat on the lake (in mph).
Time taken when traveling against the current: 22.5 / (b - 6)
Time taken when traveling with the current: 22.5 / (b + 6)
Since the total travel time is 9 hours, we can write the equation:
22.5 / (b - 6) + 22.5 / (b + 6) = 9
To solve this equation, you can cross-multiply and simplify it, or use a graphing calculator or computer algebra system. The resulting value of b will be the speed of the boat on the lake that satisfies the ferry operator's needs.
2. Let's denote the two consecutive positive integers as x and x + 1.
We are given that the sum of their reciprocals is 17/72, so we can set up the equation:
1/x + 1/(x + 1) = 17/72
To solve this equation, we can cross-multiply and simplify:
72(x + 1) + 72x = 17x(x + 1)
Expand and rearrange the equation:
72x + 72 + 72x = 17x^2 + 17x
Combine like terms:
144x + 72 = 17x^2 + 17x
Rearrange to bring all terms to one side:
17x^2 - 127x - 72 = 0
Now, you can solve this quadratic equation using factoring, completing the square, or the quadratic formula. The solutions you find will give you the two consecutive positive integers.
3. Let's simplify the given quotient:
a. [(x - 4)^2] / [(x + 3)(x + 1)]
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator:
[(x - 4)^2] * [(x - 3)(x - 1)] / [(x + 3)(x + 1)] * [(x - 3)(x - 1)]
This simplifies to:
(x - 4)(x - 4)(x - 3)(x - 1) / [(x + 3)(x + 1)(x - 3)(x - 1)]
Notice that the factors (x - 3) and (x - 1) appear in both the numerator and denominator. We can cancel them out:
(x - 4)(x - 4) / (x + 3)(x + 1)
The simplified form of the quotient is:
(x - 4)^2 / (x + 3)(x + 1)
The restriction on the variable x is given as x cannot equal -3 or -1, which means these values must be excluded from the domain of x.
b. The question states that the restriction on the variable x is x cannot equal -4, -3, -2, -1, or 4. Based on this restriction, the simplified form of the quotient will be the same as in part a:
(x - 4)^2 / (x + 3)(x + 1)
However, we also need to note that x cannot equal -4 in this case.