To determine how far up the side of the house the ladder will reach after Jace pulls it out 7 feet from the base, we can use the Pythagorean theorem.
Initially, we have a right triangle formed by:
- The height the ladder reaches up the house (48 feet).
- The distance from the base of the ladder to the wall of the house (which we can find using the Pythagorean theorem).
The ladder itself is the hypotenuse of this triangle, and its length is 52 feet. We can find the initial distance \( x \) from the wall to the base of the ladder by using the equation:
\[ x^2 + 48^2 = 52^2 \]
Calculating \( 48^2 \) and \( 52^2 \):
\[ 48^2 = 2304 \] \[ 52^2 = 2704 \]
Now, substituting these values into the equation:
\[ x^2 + 2304 = 2704 \]
Subtracting 2304 from both sides:
\[ x^2 = 2704 - 2304 = 400 \]
Now, taking the square root of both sides:
\[ x = \sqrt{400} = 20 \text{ feet} \]
So initially, the base of the ladder is 20 feet away from the house.
Jace pulls the ladder out 7 feet farther, making the new distance from the wall:
\[ 20 + 7 = 27 \text{ feet} \]
Now, we want to find the new height \( h \) the ladder reaches after being pulled out. We again use the Pythagorean theorem:
\[ 27^2 + h^2 = 52^2 \]
Calculating \( 27^2 \):
\[ 27^2 = 729 \]
Substituting in:
\[ 729 + h^2 = 2704 \]
Subtracting 729 from both sides:
\[ h^2 = 2704 - 729 = 1975 \]
Taking the square root gives us:
\[ h = \sqrt{1975} \approx 44.4 \text{ feet} \]
Thus, after pulling the ladder out 7 feet, it will reach approximately 44.4 feet up the side of the house.
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