1.)The difference of two natural number is 6, and the diffrence of their reciprocals is 1/36. Find the two numbers.
2.)In a river, ben found out that he could go 16miles downstream and then 4miles back upstream in a total of 48mins. What was the speed of Ben's boat if the current was 15miles per hour.
1.) To solve this problem, let's represent the two natural numbers as x and y. We are given that the difference of the two numbers is 6, which can be written as:
x - y = 6 (equation 1)
We are also given that the difference of their reciprocals is 1/36, which can be written as:
1/x - 1/y = 1/36 (equation 2)
Let's solve these equations simultaneously to find the values of x and y.
To eliminate the denominators in equation 2, multiply both sides of the equation by 36xy:
36y - 36x = xy (equation 3)
Now let's rearrange equation 1 to solve for x:
x = y + 6
Substitute this expression for x into equation 3:
36y - 36(y + 6) = y(y + 6)
Simplify the equation:
36y - 36y - 216 = y^2 + 6y
Combine like terms:
y^2 + 6y - 216 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's factor the quadratic equation:
(y - 12)(y + 18) = 0
Setting each factor to zero:
y - 12 = 0 --> y = 12
or
y + 18 = 0 --> y = -18
Since we are dealing with natural numbers, we discard the negative value. Therefore, y = 12.
Now substitute this value of y back into equation 1 to solve for x:
x - 12 = 6
x = 6 + 12
x = 18
So, the two natural numbers that satisfy the given conditions are 12 and 18.
2.) To solve this problem, let's assume the speed of Ben's boat in still water is B mph, and the speed of the river current is C mph.
When going downstream, Ben's effective speed will be the sum of the speed of the boat and the speed of the current:
Effective speed downstream = B + C
When going upstream, the effective speed will be the difference between the speed of the boat and the speed of the current:
Effective speed upstream = B - C
We are given that Ben can go 16 miles downstream and 4 miles upstream in a total of 48 minutes (which is 48/60 = 0.8 hours). So, we can set up the following equation:
(16 / (B + C)) + (4 / (B - C)) = 0.8
To simplify this equation, let's get rid of the denominators by multiplying through by (B + C) (B - C):
16(B - C) + 4(B + C) = 0.8(B + C)(B - C)
16B - 16C + 4B + 4C = 0.8(B^2 - C^2)
Combine like terms:
20B - 12C = 0.8B^2 - 0.8C^2
Rearrange the equation and multiply everything by 10 to get rid of the decimal:
8B^2 + 12C^2 - 20B + 12C = 0
Simplify further:
4B^2 + 6C^2 - 10B + 6C = 0
Now, let's try to factor this equation. However, it does not appear to be easily factorable. Instead, let's use the quadratic formula to solve for B:
B = (-(-10) ± √((-10)^2 - 4(4)(-10))) / (2(4))
Simplifying:
B = (10 ± √(100 + 160)) / 8
B = (10 ± √260) / 8
B ≈ (10 ± 16.12) / 8
This gives us two possible values for B:
B ≈ 3.27 or B ≈ -2.02
Since we are dealing with speed, the value of B cannot be negative. Therefore, B ≈ 3.27 mph.
So, the speed of Ben's boat in still water is approximately 3.27 miles per hour.