Calculus

[dinosaur93]

Badly need help with calculus! Kindly help me answer these questions pls 

A building is on fire. From the rooftop of the building, a citizen attach a rope to a adjacent building as a last alternative to flee the firey building. Let the path of rope be modelled by the following equation:

f(x) = 0.057x² - 1.34x + 10, 0 ≤ 0 ≤ 20

x is the distance between the rooftop to an adjacent building and f(x) being the height of a rope connecting two 2 buildings...

A 3 meter high fence is found 7 metres from the rooftop.

Question 1
How do you show that the rope passes over the fence?

Distance of the rope and the ground of the fence:
f(3) = 0.057(3)² - 1.34(3) + 10 = 6.493 metres

Fence is only 3 metres high, while the rope is 3.413 metres high    

Question 2
Find the exact height of the rope from the ground as its lowest point.


Question 3
The building is on fire. Is it safe for a citizen to drop to the ground at the lowest point of the rope if he is 1.7m tall? (If he falls more than 1m to the ground, the citizen will be injured.)

[brightsky]

okay, remember that x is the distance from the rooftop to the adjacent building. it is given that the fence is 7 metres from the roof top, which prompts you to work out the height of the fence 7 m from the roof top. so plug in x = 7, and after some button-pressing, you get f(x) = 3.413. since 3.413 > 3, hence the rope passes over the fence (in other wordsat that particular point 7 m from the roof top, the rope will hang 0.413 metres above the fence.)

for q2, we know the shape of the parabola already, so all we need to do is find where the stationary point and that will automatically give us the minimum point. deriving f(x), we get: f'(x) = 0.114x - 1.34. let this = 0 to find the stationary point.
0.114x - 1.34 = 0 --> x = 11.754. to find the height, i.e. the corresponding 'y'-value, plug this back into the original equation to get f(x) =2.1247..m

for q3, since f(x) describes the height of the rope, presumably from ground level, then yes it is safe for him to drop to the ground given that he hangs from the rope first before dropping. this is because 1 + 1.7 = 2.7 > 2.124..

[dinosaur93]

tnx dude!

[dinosaur93]

How about this one!


A cylinder is cast from 750cm³ of molten metal.

a. find the expression for the height of the cylinder in terms of r.

b. Given the surface area of the cylinder is . find the value of which gives a minimum surface area giving your answer to 2 d.p. (decimal place)

c. Hence, find the height of the cylinder with the minimum surface area giving your answer to 2 d.p.

[brightsky]

a.
V = pi*r^2*h = 750
rearranging:
h = 750/(pi*r^2)

b.
let A be surface area.
A = 2pi*r^2 + 2pi*r* 750/(pi*r^2)
dA/dr = 4pi*r - 1500/r^2
to find stationary point, let dA/dr = 0.
4pi*r - 1500/r^2 = 0
r = 4.92
since the parabola is U-shaped (coefficient in front of r^2 is >0), hence this is the x-coordinate for the MINIMUM point.

c.
just plug r = 4.92 back into the equation h = 750/(pi*r^2).

[dinosaur93]

Mixed Qs:
Kindly pls elaborate to justify your answer, thank you!

Question 1
The equation of the normal to the curve at x = 1.


Question 2
The is an angle between 2 and and , then equals _____?_____ .


Probability Qs
Question 3
Two letters are randomly selected from the words 'maths methods'. The probability that the two letter are both t is ___?___.


Cubic Functions
Question 4
A cubic function, , has a remainder of -4 when divided by (x-2) and has a remainder of 11 when divided by (x + 1). The values of a and b are ____?____. .


Question 5
The circle with equation has a centre coordinates, C and radius, r, respectively of ___?___. .

[REBORN]

Q1 - Find the point first --> (1,2)

Find dy/dx.

Sub x = 1 for dy/dx. You get 7/2. This is grad of tangent. Thus grad of normal is negative reciprocal of this.

Then use y = mx + c using pt (1,2)

[azn_dj]

Question 2, the answer is 9\pi /4
(the only angle where sin(\theta) and cos(\theta) are equal is \pi /4

Question 3:
2/12*1/11 = 2/121
(sample decreases for second alternative)

[azn_dj]

Question 4:
x^2 - 6x + 3^2 + y^2 + 2y + 1^2 -3^2 -1^2 -39 = 0

(x-3)^2 + (y+1)^2 = 49
(x-3)^2 + (y+1)^2 = 7^2
Centre is (3,-1) and the radius is 7.

[azn_dj]

Sorry, post above was actually question 5, not 4. I can't work out how to "edit" posts.

[dinosaur93]

according to the solutions.... the answer for Q3 is

I'm not sure how it came about...

Shouldn't it be =

[Gaijin]

Cubic Functions
Question 4
A cubic function, , has a remainder of -4 when divided by (x-2) and has a remainder of 11 when divided by (x + 1). The values of a and b are ____?____. .

let P=P(x)=2x^3 + ax^2 +bx +2        Using remainder theorem P(2)=-4 and P(-1)=11

P(2)=2(2)^3 + a(2)^2 +b(2) +2=-4
18+4a+2b=-4
2a+b=-11     ....A

P(-1)=2(-1)^3 + a(-1)^2 +b(-1) +2=11
-2+a+b+2=11
a-b=11        ....B

A+B                                                     Using simultaneous equations

a=0
b=-11

[Gaijin]

according to the solutions.... the answer for Q3 is

That is the correct answer

Question 3:
2/12*1/11 = 2/121
(sample decreases for second alternative)

azn_dj's method is correct but it should be:

2/12*1/11 = 2/132 =1/66

[dc302]

I actually pointed out everything that Gaijin said but somehow my posts got deleted... must have been the server problem or something.

[b^3]

I actually pointed out everything that Gaijin said but somehow my posts got deleted... must have been the server problem or something.

Yeh when the new server went down, AN went back to the database that was two days before that. I'm not sure if those two days of posts are going to be back when AN moves back to the new server (I remember someone saying they are saved somewhere), so they may reapper, don't know what order though.