You and your friend part at an intersection. You drive off north at a constant speed, and your friend drives east at a speed that is 10 mph higher. After 4 hours the distance between you and your friend is 368.78 miles. You have been driving at mph. (Round to the nearest mile).
Is your sketch a right-angled triangle with sides 4x, 4(x+10), and a hypotenuse of 368.78 ?
I see it as
(4x^2) + (4x+40)^2 = 368.78^2
use the quadratic formula to solve for x
Let's break down the problem step-by-step:
1. Start by assigning variables to the unknowns:
Let x be your speed in mph.
Your friend's speed will be x + 10 mph.
The distance traveled after 4 hours will be 4x miles for you and 4(x + 10) miles for your friend.
2. Use the Pythagorean theorem to find the distance between you and your friend:
According to the Pythagorean theorem, the square of the hypotenuse (distance between you and your friend) is equal to the sum of the squares of the other two sides.
So, we have (4x)^2 + [4(x + 10)]^2 = 368.78^2.
3. Simplify and solve the equation:
Expanding and simplifying, we get:
16x^2 + 16(x + 10)^2 = 136053.84.
Simplify the equation further and move all terms to one side:
16x^2 + 16(x^2 + 20x + 100) - 136053.84 = 0.
Simplify again:
16x^2 + 16x^2 + 320x + 1600 - 136053.84 = 0.
Combine like terms:
32x^2 + 320x - 134453.84 = 0.
4. Solve the quadratic equation:
Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 32, b = 320, and c = -134453.84.
Substituting the values into the quadratic formula:
x = (-320 ± √(320^2 - 4 * 32 * -134453.84)) / (2 * 32).
Simplifying further, we have:
x = (-320 ± √(102400 - (-17230486.88))) / 64.
x = (-320 ± √(17372986.88)) / 64.
5. Solve for x:
We can disregard the negative value since we are looking for a positive speed.
x = (-320 + √(17372986.88)) / 64.
x = (894.7705012) / 64.
x ≈ 13.98.
6. Round to the nearest mile:
The speed you have been driving at is approximately 14 mph.
Therefore, you have been driving at approximately 14 mph.
To solve this problem, we can use the Pythagorean theorem to find the distance between you and your friend. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's assume your speed is x mph. Since your friend is driving 10 mph faster, their speed would be (x + 10) mph.
We can now set up a right-angled triangle to represent the situation. The distance you have traveled after 4 hours would be 4x miles (since speed multiplied by time gives distance). Similarly, the distance your friend traveled would be 4(x + 10) miles.
We can now use the Pythagorean theorem:
distance^2 = (distance you traveled)^2 + (distance your friend traveled)^2
(368.78)^2 = (4x)^2 + (4(x + 10))^2
To solve this equation, we need to find the value of x.
(136144.6884) = (16x^2) + (16(x + 10))^2
136144.6884 = 16x^2 + 16(x^2 + 20x + 100)
136144.6884 = 16x^2 + 16x^2 + 320x + 1600
136144.6884 = 32x^2 + 320x + 1600
Rearranging the equation:
32x^2 + 320x + 1600 - 136144.6884 = 0
32x^2 + 320x - 134544.6884 = 0
Now we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 32, b = 320, and c = -134544.6884
x = (-320 ± √(320^2 - 4 * 32 * -134544.6884)) / (2 * 32)
Calculating this, we get two possible values for x: -6.98 and 186.98.
However, since speed cannot be negative, the only valid solution is x = 186.98 mph.
Therefore, rounding to the nearest mile, the answer is that you have been driving at 187 mph.