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3 years ago

What is the difference in height from the top of the pillar to the bottom of the pillar ?

Mathematics

1 answer:

stiks02 [169]3 years ago

8 0

Answer:

i think the difference in height from the top to the bottom is 8ft

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How to solve part ii and iii

iragen [17]

(i) Given that

$\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(xy) = \dfrac{7\pi}{12}$

when $x=1$ this reduces to

$\tan^{-1}(1) + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}$

$\dfrac\pi4 + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}$

$2 \tan^{-1}(y) = \dfrac\pi3$

$\tan^{-1}(y) = \dfrac\pi6$

$\tan\left(\tan^{-1}(y)\right) = \tan\left(\dfrac\pi6\right)$

$\implies \boxed{y = \dfrac1{\sqrt3}}$

(ii) Differentiate $\tan^{-1}(xy)$ implicitly with respect to $x$ . By the chain and product rules,

$\dfrac d{dx} \tan^{-1}(xy) = \dfrac1{1+(xy)^2} \times \dfrac d{dx}xy = \boxed{\dfrac{y + x\frac{dy}{dx}}{1 + x^2y^2}}$

(iii) Differentiating both sides of the given equation leads to

$\dfrac1{1+x^2} + \dfrac1{1+y^2} \dfrac{dy}{dx} + \dfrac{y + x\frac{dy}{dx}}{1+x^2y^2} = 0$

where we use the result from (ii) for the derivative of $\tan^{-1}(xy)$ .

Solve for $\frac{dy}{dx}$ :

$\dfrac1{1+x^2} + \left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} + \dfrac y{1+x^2y^2} = 0$

$\left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} = -\left(\dfrac1{1+x^2} + \dfrac y{1+x^2y^2}\right)$

$\dfrac{1+x^2y^2 + x(1+y^2)}{(1+y^2)(1+x^2y^2)} \dfrac{dy}{dx} = - \dfrac{1+x^2y^2 + y(1+x^2)}{(1+x^2)(1+x^2y^2)}$

$\implies \dfrac{dy}{dx} = - \dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2) (1 + x^2y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2) (1+x^2y^2)}$

$\implies \dfrac{dy}{dx} = -\dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2)}$

From part (i), we have $x=1$ and $y=\frac1{\sqrt3}$ , and substituting these leads to

$\dfrac{dy}{dx} = -\dfrac{\left(1 + \frac13 + \frac1{\sqrt3} + \frac1{\sqrt3}\right) \left(1 + \frac13\right)}{\left(1 + \frac13 + 1 + \frac13\right) \left(1 + 1\right)}$

$\dfrac{dy}{dx} = -\dfrac{\left(\frac43 + \frac2{\sqrt3}\right) \times \frac43}{\frac83 \times 2}$

$\dfrac{dy}{dx} = -\dfrac13 - \dfrac1{2\sqrt3}$

as required.

3 0

2 years ago

6.6666 rounded to the thousandths

NikAS [45]

The answer is 6.67 hope this helps :D

6 0

3 years ago

Read 2 more answers

Consider RS endpoints R(-2,1) and S(4,4) Find point Q that is 2/3 away NEED THIS ASAP PLEASE ANSWER

salantis [7]

<u>The question does not clearly specify from which endpoint Q is at 2/3. I'll assume Q is 2/3 away from R.</u>

Answer:

<em>The point Q is (2,3)</em>

Step-by-step explanation:

Take the aligned points R(-2,1), S(4,4), and Q(x,y) in such a way that Q is 2/3 away from R (assumed).

The required point Q must satisfy the relation:

d(RQ) = 2/3 d(RS)

Where d is the distance between two points.

The horizontal and vertical axes also satisfy the same relation:

x(RQ) = 2/3 x(RS)

$x_R-x_Q=2/3(x_R-x_S)$

And, similarly:

$y_R-y_Q=2/3(y_R-y_S)$

Working on the first condition:

$-2-x=2/3(-2-4)=2/3(-6)$

Removing the parentheses:

$-2-x=-4$

Adding 2:

$-x = -2$

x = 2

Similarly, working with the vertical component:

$1-y=2/3(1-4)=2/3(-3)$

Removing the parentheses:

$1-y=-2$

Subtracting 1:

$-y = -3$

y = 3

The point Q is (2,3)

7 0

3 years ago

Suppose you are making a scale drawing.find the length of each object on the scale drawing with the given scale then find the sc

Korvikt [17]

6.02 in long. First you need to find 6/14 then you multiply that answer by 14

4 0

3 years ago

7hr= how many minutes

denis-greek [22]

Answer:

420 minutes

Step-by-step explanation:

1 hour = 60 minutes

7 * 1 hour = 7 * 60 minutes

7 hours = 420 minutes

5 0

3 years ago

Read 2 more answers

What is the difference in height from the top of the pillar to the bottom of the pillar ?​

What is the difference in height from the top of the pillar to the bottom of the pillar ?