One leg of a right triangle is twice as long as the other. Write the length of the hypotenuse as a function of the length of the shorter leg.

The quadratic form of (x² - 1)² + (x² - 1) - 12 = 0 is (x² + 3)(x² - 4) = 0 and the roots are x = ± i√3 Or x = ± 2.

A quadratic equation is an algebraic expression in the form of variables and constants.

A quadratic equation has two roots as its degree is two.

Given, (x² - 1)² + (x² - 1) - 12 = 0.

∴ x⁴ - 2x² + 1 + x² - 1 - 12 = 0.

x⁴ - x² - 12 = 0.

x⁴ + 3x² - 4x² - 12 = 0.

x²(x² + 3) - 4(x² + 3) = 0.

(x² + 3)(x² - 4) = 0.

Now (x² + 3) = 0 Or (x² - 4) = 0.

x² = - 3 Or x² = 4.

x = ± √-3 Or x = ± 2.

x = ± i√3 Or x = ± 2.

We can also solve this equation by assuming x² = t and substitute we will get the roots in terms of t and then we'll replace x² again in t.

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The constant of proportionality on this case is 3.2

What is constant of proportionality?

The ratio between any two given values that establishes what is referred to as a proportional relationship is the constant of proportionality.

Main body:

Total amount spent by Mom = $ 72

total gallon bought = 22.5

So constant of proportionality can be found by finding value of 1 gallon of gasoline, which can be found by dividing as follows,

Price of 1 gallon = total spent/ total gasoline bought

                           = 72/22.5

                           = 3.2

Hence constant of  proportionality is 3.2.

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Step-by-step explanation:

You need to replace x in the equation with x in the table.

Isolating the variable x we will get the simplified inequality (which is equivalent to the given one)

3 < x ≤ 5.2

Sadly, the options are not given, so we can get a trivial equivalent inequality which is the one where the variable x is isolated, and it gives the solution set for the inequality.

So let's isolate the variable:

-22 > -5x - 7 ≥ −33

First, we can add 7 everywhere, so we get:

-22 + 7 > -5x - 7 + 7  ≥ −33 + 7

-15 > -5x  ≥ −26

Now we can divide everywhere by -5, because it is a negative number, it changes the direction of the symbols.

-15/-5 < -5x/-5 ≤ -26/-5

3 < x ≤ 5.2

That is the equivalent inequality, and the solution set is:

(3, 5.2]

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From the procedure, the simplest form of the radical is  5√7.

A radical can also be called a surd. The surd is an irrational number. We say that it is irrational because it can not be expressed as a terminating decimal.

Now we have to look at the radical √175. We have to think of two numbers that we can multiply to obtain 175 and one of them would be a perfect square.

If we choose the numbers 7 and 25, we can write;

√175 = √7 * √25

Thus we have in the  simplest radical form 5√7.

The expression √5/12 is already in the simplest form.

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Hope this helps you with your question.

Answer:

Step-by-step explanation:

I did it on a graphing app

Answer: Write an equation in standard form that has a y-intercept at (0,-3) and also contains the point (4,5)

Step-by-step explanation:

The tick marks on the two sides mean those two lengths are the same, which also means the angles opposite them are the same measure.  (AKA It's an isosceles triangle.)

Since the two missing angles are equal, we can solve:

    28 + x + x = 180

to find that x = 76.

The two angles are 76 degrees.

And since LM = LN, we know LN = 18 in.

The measure of angle 4 is 35 degrees and the measure of angle 5 is 81 degrees

The sum of angles on a straight line is equal to 180 degrees

Then the equation will be

∠3 + 81 +∠4 = 180  degrees

Substitute the value of ∠3 in the equation

64 + 81 + ∠4 = 180  

145 + ∠4 = 180

∠4 = 180 - 145

∠4 = 35 degrees

The sum of interior  angles of the triangle is 180 degrees

∠4 + ∠5 + 64 = 180

Substitute the values in the equation

35 + ∠5 + 64 = 180

∠5 + 99 = 180

∠5 = 180 - 99

∠5 = 81 degrees

Hence, the measure of angle 4 is 35 degrees and the measure of angle 5 is 81 degrees

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Answer:

h

Step-by-step explanation:

i did the quize good luck

The solution to the inequality −17 + 2x ≤ 19 − 4x is x ≤ k. The value of k is 6.

In mathematics, the relationship between two values that are not equal is defined by inequalities.

Therefore, the inequality can be solved as follows:

−17 + 2x ≤ 19 − 4x is  x ≤ k .

Let's solve for k

−17 + 2x ≤ 19 − 4x

subtract 2x from both sides of the inequalities

−17 + 2x ≤ 19 − 4x

−17 + 2x - 2x ≤ 19 − 4x - 2x

- 17 ≤ 19 - 6x

Subtract 19 from both sides of the equation

- 17 ≤ 19 - 6x

- 17 - 19 ≤ 19 - 19 - 6x

- 36 ≤ - 6x

divide both sides by -6

- 36 / -6 ≤ - 6x / -6

6 ≥ x

x ≤ 6

Therefore, k = 6

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The weighted average of the weighted points on the number line is -0.2.

A weighted average refers to the average of a data set that considers certain numbers as more important. Weighted averages are commonly used in statistical analysis, stock portfolios and teacher grading averages. In order to determine the weighted average of the weight points on the number line:

Hence, the average weight is -0.2.

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The half-life of this substance that weighs 32 mg and one year later, the sample weighs 25.5 mg will be 3.053 year.

The duration it takes for a given quantity to fall to half of its initial value is known as the half-life. The phrase can be used to refer to other types of decay, whether or not they are exponential, but it is most frequently used in connection with atoms going through radioactive decay. The amount of time needed for the reactant concentration to drop to half its initial value is known as the half-life of a reaction.

Here,

The initial weight of radioactive substance=32 mg

The final weight of radioactive substance=25.5 mg

Duration=1 year

The formula,

N(t)=N(0)*1/2^(t/t₁/₂)

25.5=32*1/2^(1/t₁/₂)

half-life, t₁/₂ = 3.053 year

The half-life of this substance, which weighs 32 mg, will be 3.053 years when the sample weighs 25.5 mg.

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Answer: (3,2)

Step-by-step explanation:

[tex]7x-7y=7\\\-4x-7y=-26[/tex]

subtract second equation from the first

[tex]11x=33\\x= 3[/tex]

plug x = 3 back into first equation

[tex]7(3) - 7y = 721 - 7y = 7-7y = -14\\y = -2[/tex]

Answer:

x=3 y=2

Step-by-step explanation:

The required standard equation is given as y(t) = 5.39 sin(4πt + 0.379π).

Given that,

To rewrite y(t)=2×sin 4pi t+5×cos 4pi t in the form y(t)=Asin (wt+Ф).

These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.

Here
We know that.
Asin(α + β) = Asinαcosβ + Acosαsinβ,
Comparing it with the given equation,
Asinα = 5  
Acosα = 2
Taking the ratio of the equation,
sinα/cosα = 5/2
tanα = 5/2
α = 0.379π
Now,
Asin0.379π = 5
A = 5 / sin0.379
A = 5.39

Now,
The standard form equation is given as, y(t) = 5.39 sin(4πt + 0.379π) where ω = 4π.

Thus, the required equation is given as y(t) = 5.39 sin(4πt + 0.379π).

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The area of the given triangle is 21.33 cm²

This formula is understood for its simple calculation supporting the length of three sides of a triangle, if the length of three sides are a, b, and c, then its semi-perimeter is. s=a+b+c2. Thus, the area is given by,

A=√s(s−a)(s−b)(s−c)

Given a triangle, with lengths 8 cm, 6 cm and 12 cm

Perimeter = 8+6+12 = 26 cm

Semi perimeter = 26/2 = 13 cm

Area = √s(s−a)(s−b)(s−c)

Area = √13(13-8)(13-6)(13-12)

Area = √455

Area =  21.33 cm²

Hence, The area of the given triangle is 21.33 cm²

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Answer:

450 centimeters

Step-by-step explanation:

We know that,

1 meter = 100 centimeters

so 4.5 meters = 4.5*100 centimeters

                      =450 centimeters

Answer:

  (d)  HA

Step-by-step explanation:

You have two right triangles with the hypotenuse and one angle marked as congruent. You want to know the applicable theorem for claiming the triangles are congruent.

When two triangles are known to be right triangles, the known angle is opposite the longest side (hypotenuse), and the two legs are the sides that form that known angle. Hence, some special theorems can be invoked for right triangles:

  LA — leg, angle; equivalent to ASA

  LL — leg, leg; equivalent to SAS

  HA — hypotenuse, angle; equivalent to AAS

  HL — hypotenuse, leg; no equivalent for non-right triangles

The given triangles have the hypotenuse (H) and an acute angle (A) marked, so the HA theorem can be used to prove congruence.

__

Additional comment

The HL condition described above refers two two consecutive sides and the angle not between them (SSA). This set of descriptors will create a unique triangle if and only if the angle is opposite the longest side. In general, that will not be the case, but it is guaranteed to be the case for a right triangle.

75% of the eruptions lasted between 1.77 and 4.93 minutes.

Chebyshev's Theorem calculates the proportion of observations that are within a given number of standard deviations of the mean. This theorem is applicable to many different probability distributions. Chebyshev's inequality is another name for Chebyshev's theorem.

According to Chebyshev’s theorem, the proportion of the data lying within k standard deviations of the mean is always at least 1 – 1/k^2, where k>1.

Given

Mean = 3.35

SD = 0.79

Mean – 1*SD = 3.35 – 1*0.79

= 3.35 – 0.79

= 2.56

Mean + 1*SD =3.35 + 1*0.79

= 3.35 + 0.79

= 4.14

Mean – 2*SD =3.35 – 2*0.79

= 3.35 – 1.58

= 1.77

Mean + 2*SD =3.35 + 2*0.79

= 3.35 + 1.58

= 4.93

Here, we have k = 2

For k = 2,

= 1- (1/k2)

= 1- (1/(2)2

= 1- ¼

= ¾

at least ¾ or 75% of all values lie within 2 standard deviations of the mean.

Therefore 75% of the eruptions lasted between 1.77 and 4.93 minutes.

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The distance to the top of the building changing when they are 60 feet from the base of the building is 2.19 ft/s.

In the given question we have to find how fast is the distance to the top of the building changing when they are 60 feet from the base of the building.

A family is walking toward their home, which is in an apartment building 123 feet tall.

They are walking at a rate of 5 ft/sec.

So according to the given statement diagram is

In the diagram x represent base and s represent the distance.

Using the Pythagorean theorem to find the value of s.

So the equation should be

[tex]s^2=x^2+(123)^2[/tex]........................(1)

From the question, x=60 feet

Now putting the value

[tex]s^2=(60)^2+(123)^2[/tex]

[tex]s^2[/tex] = 3600+15129

[tex]s^2[/tex] = 18729

s = 136.85

Differentiate equation 1 with respect to t using the power rule then solve the expression for ds/dt

2s ds/dt=2x dx/dt+0

2s ds/dt= 2x dx/dt

Divide by 2s on both side, we get

ds/dt= x/s dx/dt.......................(2)

They are walking at a rate of 5 ft/sec.

So dx/dt=5

Putting the value in the equation 2

ds/dt=60/136.85 ×5

ds/dt=0.438×5

ds/dt=2.19 ft/s

The distance to the top of the building changing when they are 60 feet from the base of the building is 2.19 ft/s.

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Answer:

Step-by-step explanation:

Answer:

The correct answer is D.

Step-by-step explanation:

1. You add all 4 of the given closing prices with answer choice D ($56.24)  Getting $271.35

2. Then divide the total of $271.35 by 5. which gives you $54.27.

meaning that your answer is D.

Sorry if this didn't help very much... Would appreciate a 5 star if it did :)

Answer:

i dont see image..

Step-by-step explanation:

The equivalent fraction for the given fraction -11/-4 is -11/(-4) and -11/-4.

Fraction.

Fraction means a number expressed as a quotient, in which a numerator is divided by a denominator.

Given,

Here we have the fraction -11/-4.

Now, we have to find the equivalent fraction of this fraction.

In order to find the equivalent fraction, we have to identify that the given term is reduceable or not.

If it is reduceable, then we have to simplify it to get the equivalent fraction.

But here the given fraction -11/-4 is not reduceable.

So, the next thing we have to do is to find the sign equivalent value for this one.

In the given fraction both the numerator and the denominator are in negative sign.

So, the equivalent fraction, must have both positive or both negative one.

Therefore, the equivalent fraction are -11/(-4) and -11/-4.

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Three mathematical terms with no definitions: (1) The terms (ii) and (iii) are correct., (2) The term (ii) is correct., (3) The term (i) and (ii) is correct.

What is parallel and perpendicular lines?

The definition of a parallel line is a pair of lines that never cross and always have the same distance between them. Lines that cross at right angles are referred to as perpendicular lines (90 degrees).

(1) The mathematical term parallel lines explicitly uses the undefined term(s).

The terms (ii) and (iii) are correct.

(2) The mathematical term perpendicular lines explicitly uses the undefined term(s)

The term (ii) is correct.

(3) The mathematical term line segment explicitly uses the undefined term(s).

The term (i) and (ii) is correct.

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The rigid transformation of the pre-image ΔABC, that produces the image ΔDEF is option E as follows;

E. [tex]R_{(90^{\circ},\,(0,\, 0))}[/tex]

A rigid transformation is one in which the distances between each pair of points on the pre-image is preserved on the image, such that the pre-image and the image have the same size.

The coordinates of the vertices of ΔABC are;

A(-3, -3), B(-1, -4), C(-3, -4)

The coordinates of the vertices of ΔDEF are; D(-3, 3), E(-4, 1), F(-4, 3)

The formula for the rotation of a point (x, y) 90° clockwise about the origin is presented as follows;

Pre-image point [tex]{}[/tex] (Rotation 90° clockwise)  Image point after rotation

(x, y)            [tex]{}[/tex]                        ⇒               [tex]{}[/tex]           (-y, x)

The points in the image ΔDEF of pre-image triangle ΔABC which corresponds to points on ΔABC are;

Vertex A in triangle ΔABC corresponds to vertex D on triangle ΔDEF

Vertex B in triangle ΔABC corresponds to vertex E on triangle ΔDEF

Vertex C in triangle ΔABC corresponds to vertex F on triangle ΔDEF

The transformation of (x, y) ⇒ (-y, x), (which is the transformation of a 90° clockwise rotation about the origin), on the vertices of ΔABC are;

A(-3, -3)    [tex]\underset \longrightarrow {R_{(90^{\circ}, (0, 0)}}[/tex]  D(-3, 3)

B(-1, -4)    [tex]\underset \longrightarrow {R_{(90^{\circ}, (0, 0)}}[/tex]  D(-4, 1)

C(-3, -4)    [tex]\underset \longrightarrow {R_{(90^{\circ}, (0, 0)}}[/tex]  F(-4, 3)

Therefore, the triangle ΔDEF can be obtained from the triangle ΔABC by a rotation of 90° clockwise about the origin.

The correct option is; E [tex]R_{(90^{\circ}, (0, \,0))}[/tex]

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Answer:

y - 6 = 8(x + 4)

Step-by-step explanation:

The point-slope form is [tex]y-y_{1}=m(x-x_{1})[/tex], where y1 and x1 are any point on the line and m is the slope.

The formula for slope is [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

By plugging in the points we have, we get:  [tex]\frac{22-6}{-2-(-4)}=8=m[/tex]

We can use any of the two pairs of points to represent y1 and x1 so if we choose (-4,6), we get:

[tex]y-6=8(x-(-4))\\y-6=8(x+4)[/tex]

Answer: D. Y-2 = -7/4(X+9)

Answer:

A

Step-by-step explanation:

If Nita scored 62% of the marks then the maximum marks of the examination was 600  .

In the question ,

it is given that,

the number of marks scored by Nita in the examination = 372 marks

percent of marks scored by Nita in the examination = 62% ,

let the maximum marks of the examination = "x" ,

So , according to the questions

372 = 62% of x

372 = 0.62 * x       ....because 62% = 0.62

x = 372/0.62

x = 600

Therefore , If Nita scored 62% of the marks then the maximum marks of the examination was 600  .

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