one ladder is 7.2 feet longer than another. each is placed on level ground with its lower ends 8.3 ft from the the foot of a vertical wall. the longer ladder reaches 9.1 ft farther up the wall. how long is the longer ladder?
If the longer ladder is x, and reaches up y feet on the wall, then
8.3^2 + y^2 = x^2
8.3^2 + (y-9.1)^2 = (x-7.2)^2
x = 17.9
To find the length of the longer ladder, we can use a simple approach:
Let's assume that the length of the shorter ladder is X feet.
According to the problem, the longer ladder is 7.2 feet longer than the shorter ladder. Therefore, the length of the longer ladder can be expressed as X + 7.2.
Both ladders are placed on level ground with their lower ends 8.3 ft from the foot of a vertical wall. This means that we can form a right-angled triangle with the ladder, the wall, and the ground.
The difference in vertical distance is given as 9.1 ft. This implies that the height reached by the longer ladder is 9.1 ft more than the height reached by the shorter ladder.
Using the Pythagorean theorem, the relation can be expressed as:
X^2 + 8.3^2 = (X + 7.2)^2 + 9.1^2
Now, we can solve this equation to find the value of X, which represents the length of the shorter ladder. Once we have X, we can easily find the length of the longer ladder by adding 7.2 to it.
Expanding the equation:
X^2 + 68.89 = X^2 + 14.4X + 51.84 + 82.81
Simplifying the equation:
X^2 - X^2 + 14.4X - X + 68.89 - 51.84 - 82.81 = 0
Combining like terms:
13.4X - 65.76 = 0
Now, we isolate X:
13.4X = 65.76
Dividing both sides by 13.4:
X = 65.76 / 13.4
Calculating:
X ≈ 4.91
So, the length of the shorter ladder is approximately 4.91 feet.
To find the length of the longer ladder, we add the 7.2 feet:
Length of the longer ladder = 4.91 + 7.2
Thus, the longer ladder is approximately 12.11 feet long.