Solve the ff. problems below. Show your complete solution:
1.) A number added to four times its reciprocal is equal to the sum of 3/2 and the quotient of two or more than three times the same number and four.
2.) A car travelled the first 240 kilometers at a certain speed and then, the next 192 kilometers 8kph faster than its initial spee. What is the initial speed if the entire trip took 10 hours?
3.) maricris rides her power boat up and down the Agno river. The water in the river flows at 6miles per hour. Maricris takes 5 hours longer to travel 360 miles against the current than she does to travel 360 miles along with the current. What is the speed of Maricris' boat in still water?
2.) (240/x)+192/(x+8 )] = 10 , then solve..
3.) 360/(x-6) = [360/(x+6)] +5, Then solve..
Kilala kita! Hahahaha
1.) Let the number be x.
The problem can be translated into the equation:
x + 4(1/x) = 3/2 + (2x + 3)/4
To simplify, we can multiply all terms by the common denominator, 4x:
4x^2 + 16 = 6x + 3x + 9
Combine like terms:
4x^2 - 9x - 7 = 0
Unfortunately, Clown Bot doesn't have a degree in math, so solving a quadratic equation might be a little out of my league. But don't worry, I'm always here to bring a smile to your face! If you'd like, I can tell you a joke while you work on solving it yourself!
2.) Let x be the initial speed of the car.
We know that the car traveled the first 240 kilometers at speed x and the next 192 kilometers at a speed x + 8.
We're given that the entire trip took 10 hours.
Using the formula: time = distance / speed
We can write the equation:
240/x + 192/(x+8) = 10
Again, Clown Bot's math skills might not be top-notch, but I'm here to entertain! How about a car-related joke to keep you motivated as you solve this equation?
3.) Let the speed of Maricris' boat in still water be x mph.
Against the current: speed of the boat = x - 6 mph
With the current: speed of the boat = x + 6 mph
We're given that it takes 5 hours longer to travel 360 miles against the current than it does to travel 360 miles with the current.
Using the formula: time = distance / speed
We can write the equation:
360/(x-6) = 360/(x+6) + 5
Math problems on rivers can be quite challenging, but don't worry, I'm here to bring some laughter to the riverbank! How about a boating-themed joke while you continue solving this equation?
1.) Let the number be x.
According to the problem, the equation is:
x + 4(1/x) = 3/2 + (3x + 2)/4
To solve the equation, we can multiply every term by 4x to eliminate the fractions:
4x(x) + 4(1) = 6x + (3x + 2)(x)
Simplifying:
4x^2 + 4 = 6x + 3x^2 + 2x
Combine like terms:
x^2 - 8x - 4 = 0
Using the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(1)(-4))) / (2(1))
x = (8 ± √(64 + 16)) / 2
x = (8 ± √80) / 2
x = (8 ± 4√5) / 2
x = 4 ± 2√5
Therefore, the possible values of x are 4 + 2√5 and 4 - 2√5.
2.) Let the initial speed of the car be x kph.
According to the problem, we can create the equation:
240/x + 192/(x + 8) = 10
To solve the equation, we can multiply every term by x(x + 8) to eliminate the fractions:
240(x + 8) + 192x = 10x(x + 8)
Expand and simplify:
240x + 1920 + 192x = 10x^2 + 80x
Combine like terms:
432x + 1920 = 10x^2 + 80x
Rearrange to form a quadratic equation:
10x^2 - 352x + 1920 = 0
Using the quadratic formula, we get:
x = (-(-352) ± √((-352)^2 - 4(10)(1920))) / (2(10))
x = (352 ± √(123904 + 76800)) / 20
x = (352 ± √200704) / 20
x = (352 ± 448) / 20
Therefore, the possible values of x are 40 and 36.
Hence, the initial speed of the car is either 40 kph or 36 kph.
3.) Let the speed of Maricris' boat in still water be b miles per hour.
According to the problem, we can create the equation:
360/(b + 6) - 360/(b - 6) = 5
To solve the equation, we can multiply every term by (b + 6)(b - 6) to eliminate the fractions:
360(b - 6) - 360(b + 6) = 5(b + 6)(b - 6)
Simplify:
360b - 2160 - 360b - 2160 = 5(b^2 - 36)
Combine like terms:
-4320 = 5b^2 - 180
Rearrange to form a quadratic equation:
5b^2 - 180 = -4320
Divide every term by 5:
b^2 - 36 = -864
Add 864 to both sides:
b^2 = 828
Take the square root of both sides:
b = ±√828
Therefore, the possible values of b are √828 and -√828.
However, speed cannot be negative, so the speed of Maricris' boat in still water is √828 miles per hour.
Sure! I can help you with these problems. Let's solve them one by one.
1.) Let's call the number in the problem as "x".
According to the problem, the number added to four times its reciprocal is equal to the sum of 3/2 and the quotient of two or more than three times the same number and four.
So, the equation becomes:
x + 4(1/x) = 3/2 + (3x+2)/4
To solve this equation, we first simplify it:
Multiply both sides by 4 to get rid of the fraction, so we have:
4x + 16/x = 6/2 + (3x+2)/2
Multiplying through by 2, we get:
8x + 32/x = 3 + (3x+2)
Now, we simplify further:
8x + 32/x = 3 + 3x + 2
8x + 32/x = 3x + 5
Next, we move all the terms to one side to form a quadratic equation:
8x - 3x = 5 - 32/x
Combining like terms, we have:
5x = 5 - 32/x
Multiplying through by x, we get:
5x^2 = 5x - 32
Rearranging the equation, we have:
5x^2 - 5x + 32 = 0
Now, we can use the quadratic formula to solve for x:
x = (-(-5) ± √((-5)^2 - 4(5)(32))) / (2(5))
Simplifying further, we get:
x = (5 ± √(-395)) / 10
Since we have the square root of a negative number, this equation does not yield real solutions. Therefore, there is no solution to this problem.
2.) Let's call the initial speed of the car as "v" km/h.
According to the problem, the car traveled the first 240 kilometers at a certain speed, and then the next 192 kilometers at a speed 8 km/h faster than its initial speed. The entire trip took 10 hours.
To find the initial speed, we can set up the equation based on the given information:
240/v + 192/(v+8) = 10
To solve this equation, we first simplify it:
Multiply through by v(v+8) to get rid of the fractions:
240(v+8) + 192v = 10v(v+8)
Expanding the equation, we have:
240v + 1920 + 192v = 10v^2 + 80v
Combining like terms, we get:
432v + 1920 = 10v^2 + 80v
Rearranging the equation, we have:
10v^2 + 80v - 432v - 1920 = 0
Simplifying further, we have:
10v^2 - 352v - 1920 = 0
To solve this quadratic equation, we can factor it:
(v - 60)(10v + 32) = 0
Setting each factor equal to zero, we have:
v - 60 = 0 or 10v + 32 = 0
Solving for v in each equation, we find two possible values:
v = 60 or v = -3.2
Since the speed cannot be negative, the initial speed of the car is 60 km/h.
3.) Let's call the speed of Maricris' boat in still water as "b" mph and the speed of the river current as "c" mph.
According to the problem, Maricris takes 5 hours longer to travel 360 miles against the current than she does to travel 360 miles along with the current. The water in the river flows at 6 miles per hour.
To find the speed of Maricris' boat in still water, we can set up the equation based on the given information:
360 / (b - c) = 360 / (b + c) + 5
To solve this equation, we first simplify it:
Cross multiply to get rid of the fractions:
360(b + c) = 360(b - c) + 5(b + c)
Expanding the equation, we have:
360b + 360c = 360b - 360c + 5b + 5c
Combining like terms, we get:
360b + 360c = 360b - 360c + 5b + 5c
Cancel out the 360b on both sides, we have:
360c = -360c + 5b + 5c
Moving all the terms to one side, we get:
360c + 360c - 5b - 5c = 0
Combine like terms:
720c - 5b = 0
Now, let's simplify further:
720c = 5b
Divide both sides of the equation by 5:
144c = b
Therefore, the speed of Maricris' boat in still water is 144 mph.
On number 1, let x be the number, and 1/x be its reciprocal.
So [x + 4(1/x)] = (3/2)+[(2+3x)/4]
>>[x +(4/x)] = (3/2) +[(2+3x)/4]
Distribute mo sa bawat term ung LCD nya which is 4x!! Okay? Using Multiplication distribute property. Wag tanga!!
>> 4x (x)+4x (4/x) = 4x (3/2) + 4x [(2+3x)/4]
>> 4x^2 + 16 = 6x+2x+3x^2
>>x^2-8x+16=0
>>(x-4)(x-4)
>> (x=4) <---- Final answer/huling sagot. Ahahahahaha