A 25ft ladder leans up against the side of a house, with the base of the ladder a distance 3ft from the wall.
If the ladder is moved out by 5ft, how far down the wall will the top of the ladder move?
a^2 + 3^2 = 25^2
a^2 + 9 = 625
a^2 = 616
a = 24.82
a^2 + 8^2 = 25^2
a^2 + 64 = 625
a^2 = 561
a = 23.69
Hmm, well, if I understand correctly, you're trying to figure out how the position of the top of the ladder changes when the ladder is moved out. Let me think about this...
Ah, I've got it! So we can think of the ladder, the distance from the base to the wall, and the distance from the top of the ladder to the ground as forming a right triangle. The ladder is the hypotenuse, the distance from the base to the wall is one of the sides, and the distance from the top of the ladder to the ground is the other side.
Now, the Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. So, we can use this to solve the problem.
The original right triangle has one side with length 3ft (the distance from the base to the wall) and a hypotenuse with length 25ft (the ladder). Using the Pythagorean theorem, we can find the length of the other side, which represents the distance from the top of the ladder to the ground.
Now, if we move the ladder out by 5ft, the length of the hypotenuse remains the same (25ft), but the length of the side representing the distance from the base to the wall changes to 8ft (3ft + 5ft).
Using the same logic, we can reapply the Pythagorean theorem to find the new length of the other side, which represents the distance from the top of the ladder to the ground. By plugging in the values into the equation, we get:
(8ft)^2 = (x)^2 + (25ft)^2
Simplifying, we have:
64ft^2 = x^2 + 625ft^2
Subtracting 625ft^2 from both sides gives us:
64ft^2 - 625ft^2 = x^2
Combining like terms, we get:
-561ft^2 = x^2
Taking the square root of both sides, we find:
x ≈ √(-561ft^2)
Now, my dear human friend, I must apologize for my long and mathematical-winded explanation, but the fact is that the square root of a negative number is not a real number. Therefore, the top of the ladder doesn't move down the wall when it is moved out by 5ft since there is no value for x that satisfies the equation.
So, in summary, when the ladder is moved out by 5ft, the distance down the wall at the top of the ladder remains the same. I hope this helps!
To find out how far down the wall the top of the ladder moves, you can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (ladder) is equal to the sum of the squares of the other two sides.
In this case, the base of the ladder is 3ft away from the wall, and the ladder itself is 25ft long. We want to find out how far down the wall the ladder will move if the base is moved out by 5ft.
Let's denote the distance down the wall that the top of the ladder moves as y.
Using the Pythagorean theorem, we have:
(3 + 5)^2 + y^2 = 25^2
Simplifying the equation:
8^2 + y^2 = 25^2
64 + y^2 = 625
Subtracting 64 from both sides:
y^2 = 625 - 64
y^2 = 561
Taking the square root of both sides:
y = √561
Thus, when the ladder is moved out by 5ft, the top of the ladder will move down the wall by approximately √561 ft.
To solve this problem, you can use the properties of similar triangles. Let me explain step by step how to find the distance the top of the ladder moves down the wall when it is moved out by 5ft.
1. Start by drawing a diagram of the given situation. Draw a vertical line to represent the wall of the house, and draw a horizontal line at the bottom to represent the ground. Then, draw a slanted line to represent the ladder leaning against the wall. Label the length of the ladder as 25ft and the distance of the base of the ladder from the wall as 3ft.
2. Since the ladder is leaning against the wall, we have a right triangle formed by the wall, the ground, and the ladder. The length of the ladder is the hypotenuse of this triangle.
3. To find how far down the wall the top of the ladder moves when it is moved out by 5ft, we need to determine the length of the height of the triangle. Let's call it "h". We can use similar triangles to find this value.
4. When the ladder is moved out by 5ft, the base of the ladder will be at a distance of (3ft + 5ft = 8ft) from the wall. This new distance between the base of the ladder and the wall will be the base of our new triangle.
5. Now, we have two similar triangles where the corresponding sides are proportional. The ratio of the lengths of the corresponding sides of the two triangles will be equal.
6. Using the proportion of corresponding sides, we can set up the equation: (25ft / h) = (8ft / (h + x)), where x is the distance the top of the ladder moves down the wall.
7. Cross-multiplying, we get: 25ft * (h + x) = 8ft * h.
8. Expanding the equation, we have: 25ft * h + 25ft * x = 8ft * h.
9. Subtracting 25ft * h from both sides of the equation, we get: 25ft * x = 8ft * h - 25ft * h.
10. Simplifying the equation, we have: 25ft * x = 8ft * (h - 25ft).
11. Dividing both sides of the equation by 25ft, we get: x = (8ft * (h - 25ft)) / 25ft.
12. Simplifying further, we have: x = 8ft * (h - 25ft) / 25ft.
13. Canceling out the units of feet, we have: x = 8 * (h - 25) / 25.
14. Now, we can solve for x by substituting the value of h into the equation.
15. To find the value of h, we can use the Pythagorean theorem since we have a right triangle formed by the wall, ground, and ladder. The theorem states that in a right triangle, the square of the length of the hypotenuse (25ft^2) is equal to the sum of the squares of the lengths of the other two sides (3ft^2 + h^2).
16. Setting up the equation, we have: 25ft^2 = 3ft^2 + h^2.
17. Subtracting 3ft^2 from both sides of the equation, we get: 25ft^2 - 3ft^2 = h^2.
18. Simplifying the equation, we have: 625ft^2 - 9ft^2 = h^2.
19. Taking the square root of both sides of the equation, we get: sqrt(625ft^2 - 9ft^2) = h.
20. Calculating the square root on the right side of the equation, we have: h ≈ sqrt(616) ≈ 24.8ft (rounded to one decimal place).
21. Now, we can substitute the value of h (approximately 24.8ft) into the equation x = 8 * (h - 25) / 25 to find the distance the top of the ladder moves down the wall.
22. Plugging in the value of h, we have: x = 8 * (24.8ft - 25ft) / 25.
23. Simplifying the equation, we have: x = -8 * 0.2ft / 25.
24. Finally, evaluating the expression, we have: x = -0.16ft.
Therefore, when the ladder is moved out by 5ft, the top of the ladder will move approximately 0.16ft (or about 0.2 inches) down the wall.