Sliding ladder leaning against wall, and a triangle of maximum area

[punjabi_monster]

Hello,
Below are two questions i am unsure of. Can someone please verify my answers and tell me if i am doing something wrong. thank-you for your help.

1. A rigid beam 30 m long is leaning against a vertical wall. If the bottom of the beam is pulled horizontally away from the wall at 3 m/s, how fast is the angle between the beam and the ground changing when the bottom of the beam is 18 m from the wall?

sinθ = y/30

cosθ = 18/30
θ= 53.13

A= bh/2
A= [(x)(30sinθ)] / 2
A'= [(30sinθ) (dx/dt) + (x)(30cosθ) (dθ/dt)] / 2
0= [30sin(53.13) (3) + (18)(30cos53.13) (dθ/dt)] / 2
dθ/dt= -6.67 m/s

2.Triangle ABC is inscribed in a semicircle with diameter BC=12 cm. Find the value of angle B that produces the triangle of maximum area.
Hint: An angle inscribed in a semi-circle is a right triangle.

A=bh/2
A=(12cosB)(12sinB)
=72(2cosBsinB)
=72(sin2B)
A'=72(cos2B)
0=72cos2B
cos2B=0
2B=90°
B=45°

i got sin2B using the trig double angle formula sin2x=2sinxcosx

 

[dextercioby]

There's a very elegant approach for the second problem. No calculus required (!) :

$$A_{\mbox{triangle}} =\frac{AB\cdot AC}{2}$$ (1)

Constraint, following Pythagora's theorem

$$AB^{2}+AC^{2}=12^{2}$$ (2)

(2) can be written

$$\left(AB-AC\right)^{2} =12^{2}-2 AB\cdot AC$$ (2')

U impose that the product (which by (1)is ~Area) be maximum, therefore the RHS would have to be minimum. Since the LHS can be no less than 0, one gets that it has to be 0.

$$AB=AC \Rightarrow \hat{CBA} =45 \mbox{deg}$$

Daniel.

 

[punjabi_monster]

thanks...but is the first question right?

 

[dextercioby]

The reasoning is okay. The area of the triangle is indeed constant. I didn't check the numbers. I hope they're okay.

Daniel.

 

[Jameson]

=72(sin2B)
A'=72(cos2B)
0=72cos2B

You forgot the chain rule here, but in the end it didn't affect your answer.

$$A = 72\sin{(2B)}$$
$$A' = 72\cos{(2B)}*2$$

 

[HallsofIvy]

The reasoning is okay. The area of the triangle is indeed constant. I didn't check the numbers. I hope they're okay.

Daniel.

Are you referring to problem 1? Why would the area be constant? The area goes to 0 as the beam falls to the floor. What is definitely constant is the length of the beam- the length of the hypotenuse: Pythagorean theorem: x²+ y²= 900 where x is the distance of the bottom of the beam from the wall and y is the height of the other end up the wall. Differentiating with respect to t,
2xx'+ 2yy'= 0. At the given instant, x= 18 so y= $$\sqrt{900-18^2}= \sqrt{576}= 24. 18(3)+ 24(y')= 0 so y'= -54/24= -2.25. y= 30 sin(θ) so <br /> y'= 30 cos(θ)θ'. When x= 18, &theta= 53.13 degrees and <br /> cos(53.13)= 0.6000. -2.25= (30)(0.6)(θ') so θ'= (-2.25)/(18)= -.125 or -1/8 degrees per second.$$