# Nasreen is designing an obstacle course that includes two ladders leaning against a pole. In order for the ladders to fit in the obstacle course, one ladder must span

1.6 m horizontally and the other must span 5 m horizontally. She already has one
4.3-m ladder. How long must the other ladder be? The answer should be rounded to the nearest tenth.

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## clearly the 4.3m ladder must span the 1.6m distance. Its height is thus

h = √(4.3^2-1.6^2)

So, the length of the other ladder must be

√(h^2+5^2) = √(4.3^2-1.6^2+5^2) = 6.4

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## To determine the length of the other ladder, we can use the concept of similar triangles.

Let's assume the height of the pole is 'h' meters. Using the similar triangles formed by the ladders and the pole, we can set up the following proportion:

4.3 m (length of first ladder) / h (height of the pole) = 1.6 m (horizontal span of the other ladder) / x (length of the other ladder).

Simplifying the proportion:

4.3 / h = 1.6 / x

To solve for x, we can cross-multiply and solve for x:

4.3x = 1.6h

Now we need to find a relation between h and x using the other ladder's horizontal span. Another similar triangle can be formed using the other ladder and the pole, so we can set up the following proportion:

4.3 m (length of first ladder) / h (height of the pole) = 5 m (horizontal span of the other ladder) / x (length of the other ladder).

4.3 / h = 5 / x

Cross-multiplying and solving for x, we get:

4.3x = 5h

We can now set up a system of equations using the previous equations:

4.3x = 1.6h

4.3x = 5h

To solve this system of equations, we can divide the second equation by the first equation:

(4.3x) / (4.3x) = (5h) / (1.6h)

1 = 5h / 1.6h

Now we can solve for h:

1 = 5 / 1.6

h = 5 / 1.6

h ≈ 3.125

Now that we have found the height of the pole is approximately 3.125 meters, we can substitute this value into either of the previous equations to find the length of the other ladder:

4.3x = 1.6h

4.3x = 1.6 * 3.125

4.3x ≈ 5

x ≈ 5 / 4.3

x ≈ 1.163

Therefore, the length of the other ladder should be rounded to the nearest tenth, which is approximately 1.2 meters.

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## To find the length of the other ladder, we can use the concept of similar triangles. Let's assume the height of the pole is "h" meters and the length of the other ladder is "x" meters.

Now, we can form a right triangle with the ladder, the pole, and the ground. The lengths of the sides of this triangle are as follows:
- The height of the pole: h meters
- The length of the 4.3-meter ladder: 4.3 meters
- The horizontal span of the 4.3-meter ladder: 1.6 meters

Using these values, we can set up the following proportion between the sides of the similar triangles:
h/4.3 = x/1.6

Now, let's solve for "x". Cross-multiplying the proportion, we get:
h * 1.6 = 4.3 * x

Simplifying the equation, we have:
1.6h = 4.3x

Dividing both sides by 4.3, we get:
x = (1.6h)/4.3

Now we have the equation for the length of the other ladder in terms of the height of the pole. But we need to find the height.

Since the length of the other ladder is 5 meters horizontally, and the length of the 4.3-meter ladder is 1.6 meters horizontally, we have:
5/1.6 = h/4.3

Solving for "h", we can cross-multiply the proportion:
1.6h = 4.3 * 5

Simplifying the equation, we have:
1.6h = 21.5

Now, divide both sides by 1.6 to solve for "h":
h = 21.5/1.6

Calculating this value, we find:
h ≈ 13.4

Now, substitute this value of "h" into the equation for "x":
x = (1.6 * 13.4)/4.3

Calculating this value, we find:
x ≈ 5

Therefore, the length of the other ladder should be rounded to the nearest tenth, which is 5.0 meters.