Algebra II Question?

Question

I have been having trouble with these two sections of Algebra II, so if you could please explain the following problems it would help me understand what I am suppose to be doing:

IF YOU CAN PLEASE TRY TO ANSWER ALL THREE, BUT IF NOT AT LEAST ONE

1) The top of a 15-foot ladder is 3ft farther up a wall than the foot of the ladder is from the bottom of the wall. How far is the foot of the ladder from the bottom of the wall?


2) The side of a large tent is the shape of an isosceles triangle whose area is 54 ft^2 and whose base is 6 ft shorter than twice its height. Find the height and the base of the side of the tent.

3) A farmer plans to use 21 m of fencing to enclose a rectangular pen having area 55 m^2. Only three sides of the pen need fencing because part of an existing wall will from the fourth side. Find the dimensions of the pen.


NOTE: Section is called "Problem Solving Using Polynomial Equations"

Thanks.

Answer 1

Number one is easily solved with the pythagorean theorem for right triangles which states that A^2+B^2=C^2 where C is the hypotenuse (side opposite the right angle)
This is read as A Squared plus B Squared Equals C squared.

Draw a picture of the problem showing one leg call it A =3 ft up the wall
The hypotenuse is the ladder itself leaning against the wall C = 15 foot ladder
now we need to find the last side which is B

So we have 3^2+B^2=15^2
or
9+B^2 = 225

solve for B^2 to get
B^2=225-9

or

B^2 = 216

To isolate B take the square root of both sides of the equation to get

B=14.69

The base of the ladder is 14.69 feet from the wall

Answer 2

Only one problem per question please.

1) The top of a 15-foot ladder is 3ft farther up a wall than the foot of the ladder is from the bottom of the wall. How far is the foot of the ladder from the bottom of the wall?

Let
y = distance up the wall that the top of ladder is
y - 3 = distance from wall foot of ladder is

You have a right triangle
legs = y and (y - 3)
hypotenuse 15

Use the Pythagorean Theorem.

y² + (y - 3)² = 15²
y² + y² - 6y + 9 = 225
2y² - 6y - 216 = 0
y² - 3y - 108 = 0
(y - 9)(y - 12) = 0
y = 9 or 12

y is the smaller of the two so y = 9

The foot of the ladder is 9 feet from the bottom of the wall.