Final answer:
Using the properties of similar triangles, it is determined that the 18' ladder reaches 14.4' up the wall, which is 6.4' further than the 10' ladder.
Step-by-step explanation:
The student asked how much further up the wall does the 18’ ladder reach compared to the 10’ ladder, given that both ladders make the same angle with the ground, and the 10’ ladder reaches 8’ up the wall. To solve this, we can use similar triangles since both ladders and the wall form similar right-angle triangles. The ladders are the hypotenuses, the wall is one leg, and the ground is the other leg of these triangles.
For the 10’ ladder:
Let's call the distance from the wall to the base of the ladder ’x’.
The height up the wall is 8’.
For the 18’ ladder:
The height up the wall is what we want to find, let's call it ’y’.
The distance from the wall to the base of the ladder would be ’x’, the same as for the 10’ ladder, because they form the same angle with the ground and the wall, making the triangles similar.
Using the properties of similar triangles, the ratios of corresponding sides are equal:
’x/10 = x/18’ and ’8/10 = y/18’
Since we know the 10’ ladder reaches 8’ up, we can solve for ’x’:
’x = (10×x) / 18’
From the second ratio:
’8/10 = y/18’ => ’y = (18× 8) / 10 = 14.4’
Thus, the 18’ ladder reaches up to 14.4’ on the wall. The difference in height reached between the two ladders is ’14.4 - 8 = 6.4’ feet.
The 18’ ladder reaches 6.4’ further up the wall than the 10’ ladder.