Enter your answer and show all the steps that you use to solve this problem in the space provided.
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?
Let's denote the speed of the boat in still water as B mph.
Against the current, the effective speed of the boat will be B - 6 mph.
Against the current, the time taken for the ferry operator to cover 22.5 miles is:
22.5 / (B - 6).
With the current, the effective speed of the boat will be B + 6 mph.
With the current, the time taken for the ferry operator to cover 22.5 miles is:
22.5 / (B + 6).
Given that the total time for the round trip is 9 hours, the equation is:
22.5 / (B - 6) + 22.5 / (B + 6) = 9.
To solve this equation, we can multiply through by (B - 6)(B + 6) to get rid of the fractions:
22.5(B + 6) + 22.5(B - 6) = 9(B^2 - 36).
Expand and simplify:
22.5B + 135 + 22.5B - 135 = 9B^2 - 324
45B = 9B^2 - 324
9B^2 - 45B - 324 = 0
Now we can solve this quadratic equation using the quadratic formula:
B = [-(-45) ± sqrt((-45)^2 - 4*9*(-324))]/(2*9)
B = [45 ± sqrt(2025 + 11664)] / 18
B = [45 ± sqrt(13689)] / 18
B = [45 ± 117] / 18
B = (45 + 117) / 18 or B = (45 - 117) / 18
B = 162 / 18 or B = -72 / 18
B = 9 or B = -4
Since the boat speed can't be negative, the boat must go at a speed of 9 mph on the lake in order to serve the ferry operator's needs.
simplify the sum. state any restriction on the variables.
step by step work.
x-2/z+3 + 10x/x^2-9
To simplify the sum, we'll start by finding a common denominator for the fractions in the expression.
Given expression: (x - 2)/(z + 3) + 10x/(x^2 - 9)
Notice that x^2 - 9 is a difference of squares, so it can be factored:
x^2 - 9 = (x + 3)(x - 3)
The LCD (Least Common Denominator) for the two fractions is (z + 3)(x^2 - 9).
Now rewrite the expression with the common denominator:
[(x - 2)(x - 3) + 10x(z + 3)] / [(z + 3)(x^2 - 9)]
Expand the numerator:
[(x^2 - 5x + 6 + 10xz + 30x] / [(z + 3)(x^2 - 9)]
Collect like terms in the numerator:
(x^2 + 5x + 6 + 10xz) / [(z + 3)(x^2 - 9)]
Now, state any restrictions on the variables:
Since the original expression contained denominators of (z + 3) and (x^2 - 9), we must ensure that these denominators do not equal zero, as division by zero is undefined. Therefore, the restrictions are:
z ≠ -3
x ≠ ±3
The simplified expression is:
(x^2 + 5x + 6 + 10xz) / [(z + 3)(x^2 - 9)]