The top of a 15-foot ladder is 3 ft. father up a wall than the foot of the ladder is from the bottom of the wall. How far is the foot of the ladder from the bottom of the wall?
Let x be the distance of the foot from the wall. Then use the Pythagorean theorem.
x^2 + (x+3)^2 = 15^2 = 225
2x^2 +6x -216 = 0
x^2 +3x -108 = 0
That factors easily. (Think: 9x12 = 108)Take the positive root.
Let's assume that the distance from the foot of the ladder to the bottom of the wall is "x" feet.
According to the given information, the top of the ladder is 3 feet higher up the wall than the foot of the ladder. Therefore, the distance from the top of the ladder to the bottom of the wall would be "x + 3" feet.
Since the ladder is 15 feet long, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side, which is the ladder in this case) is equal to the sum of the squares of the other two sides.
In this case, the ladder is the hypotenuse, which is 15 feet. The other two sides are x feet and x + 3 feet.
Using the Pythagorean theorem: (x^2) + ((x + 3)^2) = (15^2)
Expanding the equation: x^2 + (x^2 + 6x + 9) = 225
Combining like terms: 2x^2 + 6x + 9 = 225
Moving all terms to one side to set the equation equal to 0: 2x^2 + 6x + 9 - 225 = 0
Simplifying: 2x^2 + 6x - 216 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the value of x.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = 6, and c = -216.
Substituting the values in the quadratic formula:
x = (-6 ± √(6^2 - 4 * 2 * -216)) / (2 * 2)
Simplifying further:
x = (-6 ± √(36 + 1728)) / 4
x = (-6 ± √(1764)) / 4
x = (-6 ± 42) / 4
Now we have two possible values for x:
x = (-6 + 42) / 4 or x = (-6 - 42) / 4
x = 36/4 or x = -48/4
Simplifying the fractions:
x = 9 or x = -12
Since the distance cannot be negative, we discard x = -12.
So, the foot of the ladder is 9 feet from the bottom of the wall.
To find the distance between the foot of the ladder and the bottom of the wall, we can set up a simple equation.
Let's assume the distance between the foot of the ladder and the bottom of the wall is 'x' feet.
According to the given information, the top of the ladder is 3 feet farther up the wall than the foot of the ladder is from the bottom of the wall. So, the distance between the top of the ladder and the bottom of the wall would be 'x + 3' feet.
Now, we know that the ladder is 15 feet long. The ladder forms a right triangle with the wall and the ground. The length of the ladder is the hypotenuse of this right triangle.
By using the Pythagorean theorem, we can write the equation:
x^2 + (x + 3)^2 = 15^2
Simplifying this equation will give us the value of 'x'.