Find x.
Do not round your answer
12 cm
6 cm
5 cm

Answer: a = 2√3

First method (Pythagoras theorem):

a² + b² = c²

a² + 2² = 4²

a² = 16 - 4

a = √12

a = 2√3

Second method (sine rule):

opposite/hypotenuse = sin(x)

a/4 = sin(60)

a = 4sin(60)

a = 2√3

Third method (tan rule):

opposite/adjacent = tan(x)

a/2 = tan(60)

a = 2tan(60)

a = 2√3

Answer:

2√3

Step-by-step explanation:

From inspection of the given triangle:

As we cannot be sure that ΔABC is a right triangle since it is not marked as such, use the cosine rule to find the exact length of side a.

Cosine Rule

[tex]a^2=b^2+c^2-2bc \cos A[/tex]

where a, b and c are the sides and A is the angle opposite side a

Given:

Substitute the given values into the formula and solve for a:

[tex]\implies a^2=2^2+4^2-2(2)(4) \cos 60^{\circ}[/tex]

[tex]\implies a^2=4+16-16\left(\dfrac{1}{2}\right)[/tex]

[tex]\implies a^2=20-8[/tex]

[tex]\implies a^2=12[/tex]

[tex]\implies a=\sqrt{12}[/tex]

[tex]\implies a=\sqrt{4 \cdot 3}[/tex]

[tex]\implies a=\sqrt{4}{\sqrt{3}[/tex]

[tex]\implies a=2\sqrt{3}[/tex]

Therefore, the exact length of side a is 2√3.

To find out if ΔABC is a right triangle, use Pythagoras Theorem to solve for side a:

[tex]\implies a^2+b^2=c^2[/tex]

[tex]\implies a^2+2^2=4^2[/tex]

[tex]\implies a^2+4=16[/tex]

[tex]\implies a^2=12[/tex]

[tex]\implies a=\sqrt{12}[/tex]

[tex]\implies a=2\sqrt{3}[/tex]

As the measure of side a is the same as the solution found when using the cosine rule, we can conclude that ΔABC is a right triangle.

Answer:

1, 1.5

Step-by-step explanation:

Use the rational roots theorem. Factor out until you get the maximum simplification.

0.375(x−1)(2x−3)(19x2+15x+9)

However, if you wanted to solve this equation for r, you will get 1, 1.5 using the quartic formula.

Answer: [tex]y=26^{\circ}[/tex]

Step-by-step explanation:

Using linear pairs, we can determine the two angles in the picture attached.

Thus, as angles in a triangle add to [tex]180^{\circ}[/tex], [tex]y=26^{\circ}[/tex]

Answer:

954 passengers

Step-by-step explanation:

We can use ratios to solve

15 buses                   18 buses

--------------             = ---------------

795 passengers      x passengers

Using cross products

15 * x = 18 * 795

15x = 14310

Divide each side by 15

15x/15 = 14310 /15

x =954

18 buses can carry 954 passengers

There's nothing particularly tricky about the limits of integration. The upper limit is a telescoping series converging to 2,

[tex]\displaystyle \sum_{n=1}^\infty \frac4{4n^2-1} = 2 \sum_{n=1}^\infty \left(\frac1{2n-1} - \frac1{2n+1}\right) \\\\ ~~~~~~~~ = 2 \left(\left(1-\frac13\right)+\left(\frac13-\frac15\right) + \left(\frac15 - \frac17\right) + \cdots\right)[/tex]

The lower limit reduces to 0 using the Riemann-Liouville definition of the fractional derivative. For [tex]q\in\Bbb Q[/tex], let

[tex]\displaystyle \frac{d^q}{dx^q} f(x) = \frac1{\Gamma(\lceil q\rceil-q)} \frac{d^{\lceil q\rceil}}{dx^{\lceil q\rceil}} \int_a^x (x-t)^{\lceil q\rceil-q-1} f(t) \, dt[/tex]

With [tex]a=0[/tex], [tex]q=\frac12[/tex] and [tex]\lceil q\rceil=1[/tex], it follows that

[tex]\displaystyle \frac{d^{1/2}}{dx^{1/2}} 1 = \frac1{\Gamma\left(\frac12\right)} \frac d{dx} \int_0^x (x-t)^{-1/2} \, dt = \frac1{\sqrt{\pi x}}[/tex]

Let

[tex]\displaystyle I = \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right)}{x^2-4x-1} \, dx[/tex]

Observe that

[tex]f(x) = \dfrac{2-x}{1+2x} = f^{-1}(x)[/tex]

is its own inverse, so by substituting [tex]\frac{2-x}{1+2x}\mapsto x[/tex], we get the equivalent integral

[tex]\displaystyle \int_0^2 \frac{\tan^{-1}(x)}{x^2-4x-1} \, dx[/tex]

We have the identity

[tex]\tan^{-1}(x) + \tan^{-1}\left(\dfrac{2-x}{1+2x}\right) = \tan^{-1}(2)[/tex]

so that

[tex]\displaystyle I + I = \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right) + \tan^{-1}(x)}{x^2-4x-1} \, dx[/tex]

[tex]\implies \displaystyle I = \frac{\tan^{-1}(2)}2 \int_0^2 \frac{dx}{(x-2)^2-5}[/tex]

The remaining integral is trivial,

[tex]\displaystyle \int_0^2 \frac{dx}{(x-2)^2-5} = \int_{-2}^0 \frac{dx}{x^2-5} \\\\ ~~~~~~~~ = \frac1{2\sqrt5} \int_{-2}^0 \left(\frac1{x-\sqrt5} - \frac1{x+\sqrt5}\right) \, dx \\\\ ~~~~~~~~= -\frac{\ln(2+\sqrt5)}{2\sqrt5} \\\\ ~~~~~~~~ = -\frac1{\sqrt5} \tanh^{-1}\left(\dfrac2{\sqrt5}\right)[/tex]

Then the integral we want is

[tex]I = \displaystyle \int_0^2 \frac{\tan^{-1}\left(\frac{2-x}{1+2x}\right)}{x^2-4x-1} \, dx = \boxed{-\frac1{2\sqrt5} \tan^{-1}(2) \, \tanh^{-1}\left(\dfrac2{\sqrt5}\right)} \approx -0.357395[/tex]

Answer:

Step-by-step explanation:

So given the three bits of information you can conclude a couple of things. The end behavior towards the right should be increasing and since there is only two turning points this means it was previously going down but had a turning point and starting going up, and before that it was going up and then started going down, those will be the two turning points. So on the left side it should be decreasing and the right side it should be increasing. The last peace of information tells you that at the second turning point, should be equal to 250k when it turned back up OR the y-intercept is 250k. Given all this information the answer should be the first graph in the first picture

Answer: D

Step-by-step explanation:

Answer:

6

Step-by-step explanation:

The length of AB wit the coordinate (-6, 5) and (8,5). is 14 units

A rectangle is a quadrilateral with 4 sides and angles

Taking the length of AB.

Since AB is slant line, the length of AB will be determined using the distance formula as shown;

D = √(-6-8)² +(5-5)

D = √(-14)²

D = AB = 14 units

Hence the length of AB wit the coordinate (-6, 5) and (8,5). is 14 units

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The ratio of blue paint to yellow paint is 2 : 3.

The total number of blue paint he used on the first and second day is 10.

The number of yellow paint he used on the first and second day is 15.

On the third day, the highest number of can of blue paint he can use is 2

On the third day, the highest number of can of yellow paint he can use is 3.

The ratio of blue paint to yellow paint can be determined by determining the simplest form of the paints used: 4 : 6 = 2: 3

Total blue paint cans used on the first and second day = 4 + 6 = 10

Blue paint cans remaining = 14 - 10 = 4

Total yellow paint cans used on the first and second day = 6 + 9 = 15 cans

Yellow paint cans remaining = 20 - 15 = 5 cans

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Answer: x=9

Step-by-step explanation:

Since this shape is a parallelogram, the diagonals perpendicularly bisect each other. So EH =HG

5x-3=2x+24

3x=27

x=9

Substituing x=9 to EH and HG

EH: 5(9)-3=42

HG: 2(9)+24=42

Since the angles of elevation of the top of two vertical towers is 45° & 60°, the ratio of the height of the towers is 0.58:1

Since the angles of elevation of the top of two vertical towers as seen from the middle point of the lines joining the foot of the towers are 45° & 60°.

The height of the tower, the line of sight and the ground form a right angled trangle

Let

Using trigonometric ratios, we have that

tanФ = h/d

= h/L/2

= 2h/L

So, h = LtanФ/2

= Ltan45°/2

= L/2 × 1

= L/2

Let

Using trigonometric ratios, we have that

tanФ' = h'/d

= h'/L/2

= 2h'/L

So, h' = LtanФ'/2

= Ltan60°/2

= L/2 × √3

= √3L/2

So, the ratio of the height of the towers is n = h/h'

= L/2 ÷ √3L/2

= 1/√3

= 1/1.732

= 0.577

≅ 0.58

So, the ratio of the height of the towers is 0.58:1

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Step-by-step explanation:

Evalate the expression when m=4 and m=6

The domain of f(x) is all real numbers > than it equal to 3, for f(x) = 3/x + 2 - √(x - 3).

The domain of a function f(x) is the set of all real values of x, for which real f(x) exists.

In the question, we are asked to find the domain of the function f(x) = 3/x + 2 - √(x - 3).

To find the domain of f(x), we need to check the real values of x, for which real f(x) exists.

We check each part of f(x):

For 3/x, every x gives a real value except x = 0.

For 2, every x gives a real value as it is not dependent on x.

For √(x - 3), real values exist when x - 3 ≥ 0, as negative square roots are not real.

Therefore, after assessing each term, we can say that the domain for f(x) is x - 3 ≥ 0, or x ≥ 3.

Therefore, the domain of f(x) is all real numbers > than it equal to 3, for f(x) = 3/x + 2 - √(x - 3).

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Answer: C, (-1, -5)

Step-by-step explanation:

A = 2, B = 4, C = -3

We know the graph opens up because a is positive.

This rules out A and D

We are left with B and C as options.

Let's change this into vertex form

[tex]y=a(x-h)^2[/tex]

Complete the square

[tex]2(x+1)^2-5[/tex]

now a = 2, h = 1, k = -5

The vertex is (h, k)

plug in values:

(-1, -5)

Answer:

35 years from now.

Step-by-step explanation:

The quantity of moose as a function of time is as follows:

[tex]M(t) = 3000(1.02)^{t}.[/tex]

Thus, for [tex]M < 6000:[/tex]

[tex]3000(1.02)^{t} < 6000\\\\1.02^{t} < 2\\\\t < \frac{log\,2}{log\,1.02}\\\\ t < 35.003\,\,years.[/tex]

f(x) moves 6 unit towards right side along X-axis and forms the function g(x).

Given that, the original function is

[tex]f(x)=x^2[/tex]

and now the function is

[tex]g(x) =(x-6)^2[/tex]

When f(x)=0 then we can get by calculating that,

[tex]x^2=0[/tex]

x=0

So, the f(x) function satisfied by point (0,0).

Now when g(x)=0 then by calculating that we get,

[tex](x-6)^2=0[/tex]

x-6 = 0

x = 6

So g(x) function satisfied by (6,0).

In geometrical words we can say that if we transform from f(x) to g(x) then (0,0) tranformed to (6,0).

That suggests that the function moves 6 unit right along X axis, which is towards positive X axis.

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Answer:

Step-by-step explanation:

Proportional relationship is:

We have the ordered pair (1, 5/4), which represents:

When x = 1, the y- coordinate is equal to the coefficient of proportionality.

In this case it represents a score for one correct answer.

The trigonometric identity 4cos(-5B) sin3B is equivalent to 2[sin(8B) - sin(2B)]

Since we have 4cos(-5B) sin3B

Using the trigonometric identity

sin(x + y) - sin(x - y) = 2cosxsiny

So, cosxsiny = [sin(x + y) - sin(x - y)]/2

Since we have 4cos(-5B) sin3B, comparing cos(-5B) sin3B with cosxsiny, we have

x = -5B and y = 3B

So, we have cos(-5B)sin3B = [sin(-5B + 3B) - sin(-5B - 3B)]/2

= [sin(-2B) - sin(-8B)]/2

=  [-sin(2B) - {-sin(8B)}]/2 [since sin(-2B) = -sin(2B)]

=  [-sin(2B) + sin(8B)]/2

= [sin(8B) - sin(2B)]/2

So, 4cos(-5B)sin3B = 4 × [sin(8B) - sin(2B)]/2

= 2[sin(8B) - sin(2B)]

So, the trigonometric identity 4cos(-5B) sin3B is equivalent to 2[sin(8B) - sin(2B)]

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The best option is C. Provide multiple worksheets with pictures of the basic shapes for children to colour.

The process of learning has often been described to include all but not limited to the following:

Children according to experts learn faster and easily remember what they see. Hence, providing multiple worksheets with pictures of the basic shapes for children to colour is the most cost-efficient strategy to support children in learning about shapes.

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Answer:

volume= [tex]912.9ft^{3}[/tex] (not sure how you want to round the answer. it could be rounded to 913 or left as 912.9)

Step-by-step explanation:

volume of cone: to find the height of the cone, subtract the cylinder height from the total height (H-h= cone height)

[tex]V=\pi r^2\frac{h}{3} \\V=\pi (5.6)^2(\frac{6.8}{3} )\\V=\pi (31.36)(2.2666...)\\V=71.08057\pi \\V=223.3062[/tex]

volume of cylinder:

[tex]V=\pi r^2h\\V=\pi (5.6)^2(7)\\V=\pi (31.36)(7)\\V=219.52\pi \\V=689.6424[/tex]

add them together:

[tex]V=223.3062+689.6424\\V=912.9486[/tex]

Answer:

6

Step-by-step explanation:

i took the test

The y-intercept given line equation is 5.9.

Given that, the line of the equation is y-2.1x+5.9.

We need to find the y-intercept of the line.

The standard form of slope-intercept form is y=mx+c.

Where, m=slope and c=y-intercept.

Using the slope-intercept form to find the y-intercept, we get c=5.9.

Hence, the y-intercept given line equation is 5.9.

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Answer:

[tex]\huge{\boxed{ \huge{\sf 10^{\frac{4}{5} } x^\frac{1}{5} }}}[/tex]

Explanation:

Exponent rules:

Given expression:

breakdown

rewrite expression

The resulting function after all transformations have been carried out is; -7log6(-x) - 2.

The transformations which ensued on the function given are as follows;

On this note, it follows that the resulting function upon all the Transformations is; = -7log6(-x) - 2.

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Answer:

the answer is=21

Step-by-step explanation:
u have to sum all of its sides
(length+breadth+height).

The intervals of the function on the graph are:

The function is given as:

f(x) = -2(x-3)² + 2

See attachment for the graph.

From the graph, we have;

Domain = (-∝, ∝)

Because it takes all real values as its input

Range = (-∝, 2]

Because it has a vertex of (3, 2) and the vertex is a maximum

Also, it increases on the interval (-∝, 3] and decreases on the interval [3, ∝)

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Using the spread sheet we can easily get the answers:

The statistical section helps find answers by putting the accurate formula or the graphing calculator.

ANOVA      

Source of Variation SS   df     MS         F        P-value      F crit

Between Groups 1471.33    2 735.68 11.59   0.001072834 3.7389

Within Groups 888.55 14 63.4677      

Total              2359.882 16    

The APA style refers to Times New Roman Font 12 pts.

The effect size of the result is 0.6233 medium

Part A

ANOVA      

Source of Variation SS   df     MS F        P-value F crit

Between Groups 1471.33    2 735.68 11.59   0.001072834 3.7389

Within Groups 888.55 14 63.4677      

Total        2359.882 16    

1) The null and alternative hypotheses are

H0: u1=u2=u3

Ha: Not all means are equal

2) The significance level is alpha = 0.05

3) The test statistic F = sb²/sw²

Which if H0 is true has an F distribution with ν1=2 and  v2= 14 degrees of freedom.

The computations are

Anova: Single Factor            

Groups Count Sum Average            Variance  

Column 1      6          373    62.16666667      78.96666667  

Column 2 4 324              81             60.66666667  

Column 3 7 402        57.42857143     51.95238095  

ANOVA      

Source of Variation SS   df     MS F        P-value F crit

Between Groups 1471.33    2 735.68 11.59   0.001072834 3.7389

Within Groups 888.55 14 63.46768707      

Total        2359.882353 16  

5) The critical region is F ≥ F(0.05)(2,14) = 3.74

6) Conclusion:

Since the computed value F= 3.73 does not fall in the critical region F> 3.74 so we accept H0 and conclude that there is no significant difference in the three means.

Part D: The effect size of the result is 0.6233 medium and  calculated by

η²= ss between/ ss between + ss error

η²= 1471/1471+889

η²= 0.6233

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The rate is 1.2%.

Simple interest is interest calculated on the principal portion of a loan or the original contribution to a savings account.

Given:

P = 5000

T =6 month= 1/2 year

SI= 300

We know,

SI= P*R*T/100

300= 500* R * 1 /200

60000= 500R

R= 60000/500

R= 120

R= 1.2%

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