Question

helen needs to clean the windows on the outside of her house. she has a 7-meter ladder, and she must place the base of the ladder 2 meters from the wall of the house. will the top of the ladder reach the windows that are 6 meters above the ground? why or why not?
○ yes, because the ladder will reach a height between 7 and 8 meters, which is greater than the windows height of 6 meters.
○ no, the ladder will not reach the windows, because the ladder will reach a height of 5 meters, which is less than the windows height of 6 meters.
○ yes, because the ladder will reach a height between 6 and 7

Explanation:

Step1: Identify the problem type

This is a right - triangle problem (ladder, wall, ground form a right triangle), so we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (ladder length = 7m), \(a\) is the base (2m), and \(b\) is the height (what we need to find).

Step2: Apply the Pythagorean theorem

Let the height be \(h\). Then \(h^{2}+2^{2}=7^{2}\). So \(h^{2}=7^{2}-2^{2}=49 - 4 = 45\).

Step3: Calculate the height

Take the square root of 45: \(h=\sqrt{45}\approx6.7\) meters. Since \(6\lt6.7\lt7\), the height the ladder reaches is between 6 and 7 meters, which is greater than 6 meters (the height of the windows). So the correct answer is the option (probably the third one) that states "Yes, because the ladder will reach a height between 6 and 7 meters, which is greater than the windows' height of 6 meters."

Answer:

Yes, because the ladder will reach a height between 6 and 7 meters, which is greater than the windows' height of 6 meters. (Assuming the third option is this, as the first two have errors: first says between 7 - 8 which is wrong, second says 5m which is wrong. Let's calculate: Using Pythagoras, height \( h = \sqrt{7^2 - 2^2}=\sqrt{49 - 4}=\sqrt{45}\approx 6.7\) meters, so between 6 and 7, so the correct "Yes" option with this reasoning.)