Mathematics Question Thread

[jamonwindeyer]

Hi, so this was a question in our HY exam which i found pretty confusing:

A factory assembles torches. Each torch requires one battery and one bulb. It is known that 6% of all batteries and 4% of all bulbs are defective. Find the probability that, in a torch selected at random, both the battery and the bulb are NOT defective. Give your answer in exact form.

Hey! So if 6% of batteries are defective, then there is a 94% chance of a battery being NOT defective. Similarly, there is a 96% chance of a bulb being NOT defective. If we want bulb not defective AND battery not defective, we multiply these together

[jamonwindeyer]

hi could anyone please help me answer this question? im super lost

Sure! Since \(BC=FG=x\), we know the other two sides must be what we have to add to get to 100. That is, if the sides BF and CG are \(y\):



To find the areas of the triangles, we need the side lengths besides the side from the rectangle. We do this with Pythagoras. For example, take the triangle FGH on the bottom. The hypotenuse is \(x\), so if the other sides are both \(d\) (they are isosceles remember!):



So the area of that triangle will be \(A=\frac{1}{2}bh=\frac{1}{2}d^2=\frac{1}{2}\times\frac{x^2}{2}=\frac{x^2}{4}\)

The triangle at the top, ABC, is the same area! you can find a similar expression for the triangles on the side, then add em up!

[kiiaaa]

Sure! Since \(BC=FG=x\), we know the other two sides must be what we have to add to get to 100. That is, if the sides BF and CG are \(y\):



To find the areas of the triangles, we need the side lengths besides the side from the rectangle. We do this with Pythagoras. For example, take the triangle FGH on the bottom. The hypotenuse is \(x\), so if the other sides are both \(d\) (they are isosceles remember!):



So the area of that triangle will be \(A=\frac{1}{2}bh=\frac{1}{2}d^2=\frac{1}{2}\times\frac{x^2}{2}=\frac{x^2}{4}\)

The triangle at the top, ABC, is the same area! you can find a similar expression for the triangles on the side, then add em up!

thank you very much!

would yo be able to help me in this question?
so i got my whole working correct legit like everything when i compare it to the answers but idk how when i feed the final answer for aii) i get -128/3 not 64/3 and i struggle to tell where i go wrong. i eliminated errors such as mixing up signs or not differentiating correctly but cant find the root cause of the issue =/
thank you so much

[jamonwindeyer]

thank you very much!

would yo be able to help me in this question?
so i got my whole working correct legit like everything when i compare it to the answers but idk how when i feed the final answer for aii) i get -128/3 not 64/3 and i struggle to tell where i go wrong. i eliminated errors such as mixing up signs or not differentiating correctly but cant find the root cause of the issue =/
thank you so much

Snap a picture of your working? I'll see if I can spot the error

[Sukakadonkadonk]

Hey, could someone please help me with this series and sequences question?

A gold ball is dropped from a height of one metre. Each time it hits the ground it bounces to two-thirds of its previous height.
Calculate the distance that the golf ball travels before it comes to rest.

Thanks, I don't really get the solution they gave.

[jakesilove]

Hey, could someone please help me with this series and sequences question?

A gold ball is dropped from a height of one metre. Each time it hits the ground it bounces to two-thirds of its previous height.
Calculate the distance that the golf ball travels before it comes to rest.

Thanks, I don't really get the solution they gave.

Hey! Let's think about what is physically happening. First, the ball will travel one meter to the ground.



Then, it will bounce UP 2/3 of a meter



Then, it will fall back down the same distance



Then, it will bounce UP 2/3 the distance of 2/3



Then, it falls back down again. Do you see a pattern? Eventually, the total distance will be



So, we can use the sum of a geometric series into infinity! The formula is



So, the total distance will be

[katnisschung]

is there any way to do this without substitution?

[jakesilove]

is there any way to do this without substitution?

There absolutely is! However, you're essentially just doing substitution, but without the extra steps. Recall that



In this case, notice that



So, we can make a slight adjustment to the original integral,



and just write the answer, which is



Obviously, this is actually a definite integral, and you can evaluate the limits. However, you could easily have done this using substitution (which is literally the same method, which simplified algebra) by letting u=x^2 (or x^2-1 if you're clever!)

[katnisschung]

There absolutely is! However, you're essentially just doing substitution, but without the extra steps. Recall that



In this case, notice that



So, we can make a slight adjustment to the original integral,



and just write the answer, which is



Obviously, this is actually a definite integral, and you can evaluate the limits. However, you could easily have done this using substitution (which is literally the same method, which simplified algebra) by letting u=x^2 (or x^2-1 if you're clever!)

wait... substitution is not in the 2u course correct?
also did u mean to take out the 2 instead of half...

[RuiAce]

wait... substitution is not in the 2u course correct?
also did u mean to take out the 2 instead of half...

He's only referring to substitution. He's not exactly saying it's in the 2u course.

Misread the question. Yes he intended to take the 2 out. I think he forgot that the original integrand had a 4 in it.

[jakesilove]

He's only referring to substitution. He's not exactly saying it's in the 2u course.

Misread the question. Yes he intended to take the 2 out. I think he forgot that the original integrand had a 4 in it.

Yep, definitely took out an incorrect factor!

[Fahim486]

Hi again, I'm sort of stuck on this question.
Thanks!

[jamonwindeyer]

Hi again, I'm sort of stuck on this question.
Thanks!

Hey! For the curve to be decreasing, the first derivative is negative, so:



You can draw a diagram of the parabola \(y=x^2-1\) to find that this occurs for \(-1<x<1\), so that is our answer!! Does that make sense?

[Fahim486]

Hey! For the curve to be decreasing, the first derivative is negative, so:



You can draw a diagram of the parabola \(y=x^2-1\) to find that this occurs for \(-1<x<1\), so that is our answer!! Does that make sense?

Yes now I finally understand why the answer is -1<x<1
Thanks so much for your help!!! 

[katnisschung]

what does real solutions mean?
in terms of the attached?
thanks