Ask question

Ask question

All categories

2 years ago

5

Two trees are next to each other in a clearing. The first tree is 13 feet tall and casts an 8-foot shadow. The second tree casts

a 33-foot shadow. How tall is the second tree to the nearest tenth of afoot?

1 answer:

SVEN [57.7K]2 years ago

3 0

The second tree is 59.5 feet
tall. Given Two trees are growing
in a clearing. The first tree

You might be interested in

Find the imaginary part of\[(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)^6.\]

iren [92.7K]

Answer:

The imaginary part is 0

Step-by-step explanation:

The number given is:

$x=(\cos(12)+i\sin(12)+ \cos(48)+ i\sin(48))^6$

First, we can expand this power using the binomial theorem:

$(a+b)^k=\sum_{j=0}^{k}\binom{k}{j}a^{k-j}b^{j}$

After that, we can apply De Moivre's theorem to expand each summand: $(\cos(a)+i\sin(a))^k=\cos(ka)+i\sin(ka)$

The final step is to find the common factor of i in the last expansion. Now:

$x^6=((\cos(12)+i\sin(12))+(\cos(48)+ i\sin(48)))^6$

$=\binom{6}{0}(\cos(12)+i\sin(12))^6(\cos(48)+ i\sin(48))^0+\binom{6}{1}(\cos(12)+i\sin(12))^5(\cos(48)+ i\sin(48))^1+\binom{6}{2}(\cos(12)+i\sin(12))^4(\cos(48)+ i\sin(48))^2+\binom{6}{3}(\cos(12)+i\sin(12))^3(\cos(48)+ i\sin(48))^3+\binom{6}{4}(\cos(12)+i\sin(12))^2(\cos(48)+ i\sin(48))^4+\binom{6}{5}(\cos(12)+i\sin(12))^1(\cos(48)+ i\sin(48))^5+\binom{6}{6}(\cos(12)+i\sin(12))^0(\cos(48)+ i\sin(48))^6$

$=(\cos(72)+i\sin(72))+6(\cos(60)+i\sin(60))(\cos(48)+ i\sin(48))+15(\cos(48)+i\sin(48))(\cos(96)+ i\sin(96))+20(\cos(36)+i\sin(36))(\cos(144)+ i\sin(144))+15(\cos(24)+i\sin(24))(\cos(192)+ i\sin(192))+6(\cos(12)+i\sin(12))(\cos(240)+ i\sin(240))+(\cos(288)+ i\sin(288))$

The last part is to multiply these factors and extract the imaginary part. This computation gives:

$Re x^6=\cos 72+6cos 60\cos 48-6\sin 60\sin 48+15\cos 96\cos 48-15\sin 96\sin 48+20\cos 36\cos 144-20\sin 36\sin 144+15\cos 24\cos 192-15\sin 24\sin 192+6\cos 12\cos 240-6\sin 12\sin 240+\cos 288$

$Im x^6=\sin 72+6cos 60\sin 48+6\sin 60\cos 48+15\cos 96\sin 48+15\sin 96\cos 48+20\cos 36\sin 144+20\sin 36\cos 144+15\cos 24\sin 192+15\sin 24\cos 192+6\cos 12\sin 240+6\sin 12\cos 240+\sin 288$

(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)

A calculator simplifies the imaginary part Im(x⁶) to 0

4 0

3 years ago

Evaluate. Write your answer as a fraction or whole number without exponents. 3^–4 =

AlladinOne [14]

Answer:

1/81.

Step-by-step explanation:

3^-4 = 1 /3^4

= 1/81.

5 0

4 years ago

A key point of unhappiness for the colonies was the British decision to the colonists without allowing them representation.

USPshnik [31]

True, without representation the colonies abilities to dispute in subjects such as taxes were hindered, causing their unhappiness.

7 0

3 years ago

Read 2 more answers

6.True or False.<br> A numeral, such as 22 is called a constant.

evablogger [386]

Answer:

True

Step-by-step explanation:

The term with no variable attached to it is the constant

4 0

2 years ago

Read 2 more answers

Given the midpoint and one of the endpoints of a line segment, find the other endpoint. Midpoint: (0,3) Endpoint: (6,-3)

andrey2020 [161]

Answer:

$(-6,9)$

Step-by-step explanation:

Midpoint: (0,3)

Endpoint: (6,-3)

Use the midpoint formula:

$M=(\frac{x_{1}+x_{2}}{2} ,\frac{y_{1}+y_{2}}{2})$

Since you already have the midpoint and you need an endpoint, let the unknown endpoint be (x,y). Take the midpoint formula apart:

$\frac{x_{1}+x{2}}{2}=m_{x}$

$\frac{y_{1}+y_{2}}{2} =m_{y}$

$m_{x}$ and $m_{y}$ are the coordinates of the midpoint. Enter the known values of the midpoint into the equations:

$(0_{m_{x}},3_{m_{y}})\\\\\frac{x_{1}+x_{2}}{2}=0 \\\\\frac{y_{1}+y{2}}{2}=3$

Now enter the known endpoint values:

$(6_{x_{1}},-3_{y_{1}})\\\\\frac{6+x_{2}}{2}=0\\\\\frac{-3+y_{2}}{2}=3$

Solve for x. Multiply both sides by 2:

$2*(\frac{6+x}{2})=2*(0)\\\\6+x=0$

Subtract 6 from both sides:

$6-6+x=0-6\\x=-6$

Now solve for y. Multiply both sides by 2:

$2*(\frac{-3+y}{2})=2*(3)\\\\ -3+y=6$

Add 3 to both sides:

$-3+3+y=6+3\\y=9$

Now take the values of x and y and turn into a point:

$x=-6\\y=9\\(-6,9)$

Finito.

7 0

3 years ago