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3 years ago

Solve for x in the partial spiral below.

Mathematics

2 answers:

Andrew [12]3 years ago

8 0

After further application of Pythagoras Theorem on all triangles of the spiral, we get the hypotenuses as in order: root 2, root 3, 2, root 5
Thus, the final hypotenuse, i.e., x, has the value of root 5

maks197457 [2]3 years ago

3 0

Answer:

$x = \sqrt 5$

Step-by-step explanation:

On solving by Pythagoras theorem:

(Going in counterclockwise direction)

For bottom most triangle :

Length of Hypotenuse $= \sqrt {1^2 +1^2} =\sqrt {1+1}=\sqrt 2$

The for second triangle :

Length of Hypotenuse $= \sqrt {(\sqrt 2) ^2 +1^2} =\sqrt {2+1}=\sqrt 3$

The for third triangle :

Length of Hypotenuse $= \sqrt {(\sqrt 3) ^2 +1^2} =\sqrt {3+1}=\sqrt 4 = 2$

The for fourth triangle :

x $= \sqrt {(2) ^2 +1^2} =\sqrt {4+1}=\sqrt 5$

You might be interested in

5y + 3 = 2y - 3x + 5 as a function in "y=..." form (please show work)

vlabodo [156]

Hey there!

In order to solve this equation for y, you need to use addition, subtraction, and division to combine and simplify the terms. You do this by first eliminating the terms, which you do by completing whatever will make each term equal to zero. Then, you'll repeat this same thing on the other side of the equation, like so:

5y + 3 = 2y – 3x + 5
– 3 – 3

5y = 2y – 3x + 2

As you can see, I subtracted three from +3 the left side to cancel it out, then repeated that on the right side to the similar term (the constant). Now, just repeat that with your other terms until your equation is equal to y.

5y = 2y – 3x + 2
+ 2y + 2y

7y = –3x + 2
7 7

y = –3x + 2
7

When you can't simplify any more, you have your answer. y will be equal to –3x + 2 over 7.

Hope this helped you out! :-)

4 0

4 years ago

"You are dating Moon rocks based on their proportions of uranium-238 (half-life of about 4.5 billion years) and its ultimate dec

Olenka [21]

Step-by-step explanation:

Initial mass of the isotope = x

Time taken by the sample to decay its mass by 41% = t

Formula used :

$N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}$

where,

$N_o$ = initial mass of isotope = x

N = mass of the parent isotope left after the time, (t) = 59% of x = 0.59x

$t_{\frac{1}{2}}$ = half life of the isotope = 4.5 billion years

$\lambda$ = rate constant

Now put all the given values in this formula, we get

$0.59x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}$

t = 3.4 billion years

The age a rock is 3.4 billion years.

Initial mass of the isotope = x

Time taken by the sample to decay its mass by 35%= t

Formula used :

$N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}$

where,

$N_o$ = initial mass of isotope = x

N = mass of the parent isotope left after the time, (t) = 65% of x = 0.65x

$t_{\frac{1}{2}}$ = half life of the isotope = 4.5 billion years

$\lambda$ = rate constant

Now put all the given values in this formula, we get

$0.65x=x\times e^{-(\frac{0.693}{\text{4.5 billion years}})\times t}$

t = 2.8 billion years

The age a rock is 2.8 billion years.

5 0

4 years ago

How does the graph of g(x) = <img src="https://tex.z-dn.net/?f=%28x%2B12%29%5E%7B2%7D" id="TexFormula1" title="(x+12)^{2}" alt="

Law Incorporation [45]

Answer:

Horizontal translation of the parent graph

Step-by-step explanation:

<h2>Definitions:</h2>

In the vertex form of a quadratic function, f(x) = a(x - h)² + k, where:

(h, k) = vertex of the graph

a = determines the width and direction of the graph's opening.

A horizonal translation to the parent graph is given by, y = f(x - h), where:

h > 0 ⇒ Horizontal translation of h units to the right

h < 0 ⇒ Horizontal translation of |h | units to the left

In the graph of g(x) = (x + 12)², the vertex occurs at point (-12, 0).

While the vertex of the parent graph, f(x) = x² occurs at point, (0, 0).

<h2>Answers:</h2>

Since the vertex of g(x) occurs at point, (-12, 0), substituting the value of (h, k ) into the vertex form will result into:

g(x) = a(x - h)² + k

g(x) = [x - (-12)]² + 0

g(x) = (x + 12)² + 0

g(x) = (x + 12)²

Therefore, the graph of g(x) = (x + 12)² represents the horizontal translation of the parent graph, f(x) = x², where the graph of g(x) is horizontally translated 12 units to the left.

3 0

3 years ago

Manav is painting a rectangular room and the dimensions of this room are given by x,y and z feet. Manav takes 8 hours to paint a

zhenek [66]

Answer:

B) 36 cubic feet

Step-by-step explanation:

Dimensions of the room are x, y and z feet. It is given that x = 6 feet.

Manav took 8 hours to paint the wall with dimensions x and z.

This means, in 8 hours Manav painted area = xz = 6z

In 1 hour Manav painted area = $\frac{6z}{8}$

Manav took 4 hours to paint the wall with dimensions y and z.

This means, in 4 hours Manav painted area = yz

In 1 hour Manav painted area = $\frac{yz}{4}$

Manav took 12 hours to paint the ceiling with dimensions x and y.

This means, in 12 hours Manav painted area = xy = 6y

In 1 hour Manav painted the area = $\frac{6y}{12}=\frac{y}{2}$

Since, Manav works at constant rate, this means the worked done in 1 hour in all 3 cases would be the same. So we can write:

$\frac{6z}{8}=\frac{yz}{4}=\frac{y}{2}$

Using the first two equations, we get:

$\frac{6z}{8}=\frac{yz}{4}\\\\ \frac{6}{8}=\frac{y}{4}\\\\ y=\frac{6}{8} \times 4\\\\ y = 3$

Using the last two equations, we get:

$\frac{yz}{4}=\frac{y}{2}\\\\ \frac{z}{4}=\frac{1}{2}\\\\z=2$

So, we have the follwoing dimensions of the room:

x = 6 feet

y = 3 feet

z = 2 feet

So the volume of the room is = 6 x 3 x 2 = 36 cubic feet

7 0

3 years ago

Calculate the new balance of an account that has been sitting for three years with a 1.5% interest rate and had an initial depos

maksim [4K]

Answer:

$5225

Step-by-step explanation:

Use the formula for the amount after simple interest: $A = P(1 + rt)$

"A" is the final amount, or balance.

"P" is the principal, or the starting amount.

"r" is the rate of interest in decimal form.

"t" is the time.

What we know:

P = 5000

t = 3

r = 1.5%

Convert the rate to decimal form by dividing by 100, or moving the decimal place two places to the left.

1.5% => 0.015 = r

Substitute what we know into the formula:

A = P(1 + rt)

A = 5000(1 + (0.015)(3)) <=simplify

A = 5000(1 + 0.045)

A = 5000(1.045)

A = 5225 <= new balance

The new balance of an account is $5225.

4 0

3 years ago