A cylindrical tank, 6 foot in radius, lies on it's side parallel and against the side of a warehouse. A ladder leans against the building, passes over and just touches the tank, and has a slope of -3/4. Find the equation of the ladder and the length of the ladder
near the 2/3 mark of solution should say
so P is (10.8 , 9.6)
typo had no effect on rest of solution
@Reiny I think you get wrong at the slope of equation #1, and it makes your answer after #1 is wrong also.. but I think your procedure is correct.. Thank You
Well, it seems like this ladder is quite the overachiever, touching both the building and the tank! Let's solve this mathematically and have some fun with it!
First, let's find the equation of the ladder. We know that the ladder has a slope of -3/4, so we can use the point-slope form of a linear equation to find the equation of the ladder. We'll choose a point on the ladder where it touches the tank.
The center of the tank is on the ground level and is aligned with the center of the ladder where it touches the tank. So, the coordinates of this point are (6, 0).
Using the point-slope form, y - y₁ = m(x - x₁), where (x₁, y₁) are the coordinates of the point and m is the slope, we can plug in the values: y - 0 = (-3/4)(x - 6).
Simplifying the equation, we get y = -3/4x + 9/2. So, the equation of the ladder is y = -3/4x + 9/2.
Now, let's move on to finding the length of the ladder. We have a right triangle formed by the ladder, the distance from the ground to the top of the tank (which is 6 feet), and the distance from the top of the tank to the top of the building (which is unknown).
Using the Pythagorean theorem, we can find the length of the ladder:
(a^2 + b^2 = c^2)
6^2 + b^2 = c^2
36 + b^2 = c^2
b^2 = c^2 - 36
Since b is the length of the ladder, and we want to find its value, we'll solve for b:
b = √(c^2 - 36)
Now, let's go back to the equation of the ladder (y = -3/4x + 9/2) to find the value of x when y is 6 (the height of the tank). Plug in 6 for y:
6 = -3/4x + 9/2
Multiply through by 4 to get rid of fractions:
24 = -3x + 18
Subtract 18 from both sides:
6 = -3x
Divide by -3:
x = -2
Now we know that at the height of the tank, x is -2.
Plug this value into the equation for b:
b = √((-2)^2 - 36)
b = √(4 - 36)
b = √(-32)
Uh-oh! We run into a problem here. The square root of a negative number is not a real number, so that means the ladder doesn't touch the top of the building.
Well, looks like our ladder isn't long enough to reach the top of the building. Maybe we should send in a taller clown to suggest some other funny solutions!
To find the equation of the ladder and the length of the ladder, we can use the concept of similar triangles.
Let's denote the height of the tank as 'h' and the length of the ladder as 'L'. The radius of the tank is given as 6 feet.
Since the ladder touches the tank, it can be visualized as the hypotenuse of a right triangle formed between the ladder, the distance along the ground from the point of contact to the center of the tank (which is the radius), and the height of the tank.
Given that the slope of the ladder is -3/4, we can write the equation of the ladder as:
(h - 6) / (-r) = -3/4
where r is the radius of the tank.
Simplifying the equation, we get:
(h - 6) / 6 = 3/4
Cross-multiplying, we get:
4(h - 6) = 6 * 3
4h - 24 = 18
Adding 24 to both sides, we get:
4h = 42
Dividing by 4, we get:
h = 10.5
Now that we have the height of the tank, we can find the length of the ladder using the Pythagorean theorem. The ladder, the radius, and the height of the tank form a right triangle, where the ladder is the hypotenuse.
Using the Pythagorean theorem, we have:
L^2 = r^2 + h^2
Plugging in the values, we get:
L^2 = 6^2 + 10.5^2
L^2 = 36 + 110.25
L^2 = 146.25
Taking the square root of both sides, we get:
L = √146.25
L ≈ 12.08 feet
Therefore, the equation of the ladder is (h - 6) / (-6) = -3/4, and the length of the ladder is approximately 12.08 feet.
use the equation y=mx+b
Put the diagram on the x-y grid
so the centre of the circle for the cylinder is C(6,6)
Let P(x,y) be the point of contact of the ladder with the cylinder.
Slope of ladder touching at P is -3/4
So the slope of PC = 4/3
(y-6)/(x-6) = 4/3
4(y-6) = 3(x-6)
y-6 = (3/4)(x-6) #1
equation of circle is
(x-6)^2 + (y-6)^2 = 36
but P lies on this, so
(x-6)^2 + (9/16)(x-6)^2 = 36 #2
times 16
16(x-6)^2 + 9(x-6)^2 = 576
25(x-6)^2 = 576
take √
5(x-6) = 24
x-6 = 24/5
x = 24/5 + 6 = 10.8
subbing back into #1, we get
y = 9.6
so P is (10.8 / 9.6)
equation for ladder:
let y = (-3/4)x + b
9.6 = (-3/4)(10.8) = b
b = 17.7
equation is y = -.75x + 17.7
x - intercept is 23.6
y - intercept is 17.7
length of ladder L
L^2 = 23.6^2 + 17.7^2 = 870.25
L = 29.5