a rectangle has a perimeter of 16 centimeters. what is the greatest area the rectangle can have

4(3)(2)/2 + 2² = 12 + 4 = 16

The correct equation is;

p = 4t + 1

The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of the points on the line. In general, the equation of a line can be written in slope-intercept form, which is y = mx + b, where m is the slope of the line and b is the y-intercept (the point at which the line crosses the y-axis).

We can get the slope of the graph from;

m = y2 - y1/x2 =x2 - x1

m = 1 - 0/0.25 - 0

m = 4

Since the y intercept is at y = 1 then we have;

p = 4t + 1

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The required sinusoidal function be,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

Since we know that,

The general formula of a sinusoidal function is,

⇒ y = A sin(B(x - C)) + D,

where,

A is the amplitude

B is the frequency (and related to the period by T = 2π/B)

C is the phase shift (the horizontal displacement from the origin)

D is the vertical shift (the midline)

Using the given information,

Amplitude = 25, so A = 25.

Period = 15, so T = 15.

We know that,

T = 2π/B, so we can solve for B,

⇒ 15 = 2π/B

⇒  B = 2π/15

Midline is y = 38, so D = 38.

⇒ f(1) = 63,

so we can also use this to find the phase shift:

⇒ 63 = 25 sin(B(1-C)) + 38

⇒ 25 sin(B(1-C)) = 25

⇒ sin(B(1-C)) = 1

⇒ B(1-C) = π/2 + 2nπ or 3π/2 + 2nπ,

where n is an integer.

Substituting B and solving for C in each case, we get,

⇒ B(1-C) = π/2 + 2nπ 2π/15 (1 - C)

              = π/2 + 2nπ 1 - C

              = 15/4 + 30nπ C

              = 1 - 15/4 - 30nπ

⇒ B(1-C) = 3π/2 + 2nπ 2π/15 (1 - C)

              = 3π/2 + 2nπ 1 - C

              = 15/4 + 60nπ/2 C

              = 1 - 15/4 - 30nπ

So we have two possible functions are,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

where n is any integer.

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Answer:

r=7

Step-by-step explanation:

Cylinder Area

= πr² x h

1078 = 22/7 x r² x 7

1078/22 = r²

49=r²

r=7

Answer: To obtain the relation between α and β, we can eliminate t from the given equations.

(iv)

α = 1/2a(t+1/t)

β = 1/2a(t-1/t)

We can multiply these two equations to eliminate t^2:

αβ = (1/2a(t+1/t))(1/2a(t-1/t))

αβ = (1/4a^2)(t^2 - 1/t^2)

Multiplying both sides by 4a^2 gives:

4a^2αβ = t^2 - 1/t^2

Adding 1/t^2 to both sides gives:

4a^2αβ + 1/t^2 = t^2 + 1/t^2

Multiplying both sides by t^2 gives:

4a^2αβt^2 + 1 = t^4 + 1

Rearranging and simplifying gives the relation between α and β:

4a^2αβ = t^4 - 4a^2t^2 + 1

Now we can write the locus of the point as t varies:

4a^2αβ = t^4 - 4a^2t^2 + 1

This is a fourth degree equation in t, which represents a curve in the (α, β) plane. However, we can simplify it by noting that t^2 is always non-negative. Therefore, we can treat 4a^2t^2 as a constant and write:

4a^2αβ = (t^2 - 2a^2)^2 + 1 - 4a^4

This is the equation of a conic section called a hyperbola. Its center is at (0,0), its asymptotes are the lines α = ±β, and its foci are at (a√2,0) and (-a√2,0).

Step-by-step explanation:

The correct match of each ordered pair with each function is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

To match each function with the correct ordered pair, we need to substitute the x-values from the ordered pairs into each function and see which one gives the corresponding y-value.

Substitute the x value of (-9, 74) into y = -8x + 2:

y = -8(-9) + 2

y = 74

Substitute the x value of (2,-6) into y = -4x + 2:

y = -4(2) + 2

y = -6

Substitute the x value of (9, 70) into y = 7x + 7:

y = 7(9) + 7

y = 70

Substitute the x value of (-4, 23) into y = -7x - 5:

y = -7(-4) - 5

y = 23

Therefore, the correct matching is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

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Answer:

128

Step-by-step explanation:

Answer: 128

Step-by-step explanation:

Remember the order of operations in this problem:

8² x (2 + 6) / 4

8² x (8) / 4

64 X 8 /4

512 / 4

= 128

Hope this helps!

Answer: LCM is 72. GCF is 6.

Answer:

  a)  even

  b)  4th differences: -72

  c)  minimum: 0, maximum: 4. This function has 0 real zeros.

  d)  -1377

  e)  -288

Step-by-step explanation:

You want to know a number of the characteristics of the function f(x) = -3x⁴ +6x² -10:

A function is even if f(x) = f(-x). The graph of an even function is symmetrical about the y-axis. An even polynomial function will only have terms of even degree.

The exponents of the terms of f(x) are 4, 2, 0. These are all even, so we can conclude the function is an even function.

We can also evaluate f(-x):

  f(-x) = -3(-x)⁴ +6(-x)² -10 = -3x⁴ +6x² -10 ≡ f(x) . . . . . the function is even

We can look at values of x on either side of x=0. The attachment shows function values and finite differences for x = -3, -2, ..., +3.

The fourth finite differences are constant at -72. (We expect this value to be -3·4!, the leading coefficient times the degree of the polynomial, factorial.)

A 4th-degree polynomial will always have exactly four zeros. They may be complex, rather than real. Complex zeros will come in conjugate pairs, so the number of real zeros may be 0, 2, or 4; a minimum of 0 and a maximum of 4.

This polynomial function has no real zeros. The four complex zeros are approximately ...

  ±1.18864247 ±0.64255033i

The average rate of change on the interval [a, b] is given by ...

  AROC = (f(b) -f(a))/(b -a)

For [a, b] = [2, 7], this is ...

  AROC = (((-3(7²) +6)7² -10) -((-3(2²) +6)2² -10)/(7 -2)

  = ((-147 +6)(49) -(-12 +6)(4)) / 5 = (-6909 +24)/5 = -6885/5 = -1377

The average rate of change on [2, 7] is = -1377.

The derivative of the function is ...

  f'(x) = -3(4x³) +6(2x) = 12x(-x² +1)

  f'(3) = 12·3(-3² +1) = 36(-8) = -288

The instantaneous rate of change at x=3 is -288.

Answer:

on what?

Step-by-step explanation:

How do you get rid of an inner bully?

5 Ways to Stop that Inner Bully

Become aware of what you are saying to yourself. ...

Replace this with mindful attention to your feelings. ...

Realize you are not alone in your suffering. ...

Use soothing self-talk. ...

Access Your Wise Mind.

The distance between the ship and the base of the is approximately 359.32 feet.

The distance between the ship and the base of the lighthouse can be found using the tangent of the angle of elevation.

The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (160 feet) and the adjacent side is the distance between the ship and the base of the lighthouse (x).

So, we can set up the equation:

tan(24) = 160/x

To solve for x, we can cross multiply and then divide:

x * tan(24) = 160

x = 160/tan(24)

Using a calculator, we can find that tan(24) is approximately 0.4452.

So, x = 160/0.4452

x = 359.32 feet

Therefore, the distance between the ship and the base of the lighthouse is approximately 359.32 feet.

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Answer:

y = (-5/2)x - 3

Step-by-step explanation:

To find an equation of a line that is perpendicular to the given line and passes through the point (0, -3), we need to use the fact that perpendicular lines have opposite reciprocal slopes.

The given line has a slope of 2/5, so the slope of the line perpendicular to it is:

-1 / (2/5) = -5/2

This means that the equation of the perpendicular line has the form:

y = (-5/2)x + b

where b is the y-intercept we want to find.

Since the line passes through the point (0, -3), we can substitute these values into the equation and solve for b:

-3 = (-5/2)(0) + b

b = -3

Therefore, the equation of the line perpendicular to y = 2/5x + 4 with a y-intercept of -3 is:

y = (-5/2)x - 3

polynomial expression of degree 2 with 3 terms.

a. i. \( a^{3}\left(a^{2}+a+1\right) = a^{5}+a^{4}+a^{3} \)


ii. \( \left(3 x^{2}-4 x+3\right)-(4 x-10) = 3 x^{2}-7 x-7 \)


b. i. \( a^{5}+a^{4}+a^{3} \) is a polynomial expression of degree 5 with 3 terms.


ii. \( 3 x^{2}-7 x-7 \) is a polynomial expression of degree 2 with 3 terms.

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Answer:

1-B
2-E

3-D

4-A

5-C

Step-by-step explanation:

-4x + 3y = 3

3y = 4y + 3

y = 4/3 y + 1 => slope is 4/3, y-intercept is (0,1)

Equation 1 matches with Letter B

12x - 4y = 8

4y = 12x - 8

y = 3x - 2 => slope is 3, y-intercept is (0,-2)

Equation 2 matches with Letter E

8x + 2y = 16

2y = -8x + 16

y = -4x + 8 => slope is -4, y-intercept is (0,8)

Equation 3 matches with Letter D

-x + 1/3 y = 1/3

1/3 y = x + 1/3

y = 3x + 1 => slope is 3, y-intercept is (0,1)

Equation 4 matches with Letter A

-4x + 3y = -6

3y = 4x = -6

y = 4/3 x - 2 => slope is 4/3, y-intercept is (0,-2)

Equation 5 matches with Letter C

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a. angle 2 and angle 1, angle 2 and angle 3 are linear pairs. b. angle 9 and angle 8, angle 9 and angle 5 are linear pairs. c. angle 4 and angle 2 form vertical angles.

A linear pair of angles in geometry is a pair of neighbouring angles created by the intersection of two lines. When two angles share a vertex and an arm but do not overlap, they are said to be adjacent angles. Due to their formation on a straight line, the linear pair of angles are always complementary. Thus, the total of two angles in a pair of lines is always 180 degrees.

a. angle 2 and angle 1, angle 2 and angle 3 are linear pairs.

b. angle 9 and angle 8, angle 9 and angle 5 are linear pairs.

c. angle 4 and angle 2 form vertical angles.

d. angle 8 and angle 5 form vertical angles.

e. The rays that form angle 7 and angle 9 do not for, opposite rays.

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The solution to the compound linear inequality 1.4 ≤ 9.2 - 0.8x ≤ 6.9 are the values of x in the interval [2.9, 9.8].

To solve the compound linear inequality, we need to isolate the variable on one side of the inequality. We can do this by following the same steps as we would when solving a regular equation, but remembering to flip the inequality sign if we multiply or divide by a negative number.

1.4 ≤ 9.2 - 0.8x ≤ 6.9

First, we'll subtract 9.2 from all sides of the inequality:

-7.8 ≤ -0.8x ≤ -2.3

Next, we'll divide all sides by -0.8 to isolate the variable. Remember to flip the inequality signs since we're dividing by a negative number:

9.75 ≥ x ≥ 2.875

Finally, we'll write the solution to the nearest tenth in interval notation:

[2.9, 9.8]

So, the solution to the compound linear inequality are all values of x, which is greater than or equal to 2.9 but less than or equal to 9.8, or x is in the interval [2.9, 9.8].

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Answer:

Jeremiah ate more, by 5 servings more

Step-by-step explanation:

$100 is 4 x number of vegetable servings

100/4 = 25 number of total servings

If Jeremiah ate 15 servings, his brother ate 25-15 =10

Servings Jeremiah 15, brother 10

The value of (x,y) is ( 6, -6) (optionA)

Simultaneous equations are two or more algebraic equations that share variables e.g. x and y . They are called simultaneous equations because the equations are solved at the same time. For example, below are some simultaneous equations: 2x + 4y = 14, 4x − 4y = 4. 6a + b = 18, 4a + b = 14.

6x+4y = 12 equation 1

-6x +6y = -72 equation 2

add equation 1 and 2

10y = - 60

y = -60/10

y = -6

substitute -6 for y in equation 1

6x +4(-6) = 12

6x -24 = 12

6x = 12+24

6x = 36

x = 36/6 = 6

therefore the value of (x,y) = ( 6, -6)

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Answer:

p^(2(s-t)^2)/(s+t)

Step-by-step explanation:

We can simplify this expression by using the properties of exponents:

((p^r)/(p^s))^(r+s) ((p^2)/(p^t))^(s+t) ((p^t)/(p^r))^(r+t)

= (p^(r+s-s))^r (p^(2s-2t))^s (p^(t-r+r))^t / (p^(r+s-r))^r (p^(2t-2s))^s (p^(r-t+t))^t

= p^r p^(2s-2t)s p^t / p^r p^(2t-2s)s p^t

= p^r / p^r * (p^(2s-2t))^(s/(s+t)) / (p^(2t-2s))^(s/(s+t))

= p^r / p^r * p^((2s-2t)s/(s+t)) / p^((2t-2s)s/(s+t))

= p^0 * p^(2s^2-2st-2ts+2t^2)/(s+t)

= p^(2s^2-2st-2ts+2t^2)/(s+t)

= p^(2(s-t)^2)/(s+t)

Therefore, ((p^r)/(p^s))^(r+s) ((p^2)/(p^t))^(s+t) ((p^t)/(p^r))^(r+t) simplifies to p^(2(s-t)^2)/(s+t).

Yes, the graph represents a function.

A relation is a function if it has only One y-value for each x-value.

The given ordered pairs from the given graph are (-7, 3), (-3, -3), (0,1), (2, 4), (3, -1), (5, -6)

The given graph represents a relation.

Since each value of x has unique y value.

So the given graph represents a function.

Hence, yes the graph represents a function.

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The missing side in the triangle has a length of 9.899.

In an isosceles triangle, the two sides are equal and the angles opposite to the two equal sides are also equal.

In the figure, ∠RQS = ∠RSQ =45°

Thus, it is an isosceles triangle with sides RQ=RS= 7

What is Pythagoras' theorem?

According to Pythagoras' theorem for a right-angled triangle:

Base² + Height²= Hypotensuse²

In the given figure: Base = 7 and Height = 7,

Thus, QS²= RQ² + RS²

          QS² = 7² + 7²

           QS = √98

                 =9.899    

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None of the expressions given is equivalent to (2 - ³√24)²..

Mathematical statements must contain a sentence, at least one mathematical operation, and at least two numbers or factors. With this mathematical operation, you can increase, split, add, or take something away. The shape of a phrase is as follows: Expression: (Number/Variable, , Math Operator)

Let's first simplify the expression (2 - ³√24)²:

(2 - ³√24)² = (2 - 24^(1/3))² = (2 - 2.29)² = (-0.29)² = 0.0841

Now we can check which expressions are equivalent to 0.0841 when (2 - ³√24)² is evaluated:

4 = 4.0000... (not equivalent to 0.0841)

04 = 0.04 (not equivalent to 0.0841)

26.2 - 8 = 18.2 (not equivalent to 0.0841)

02 - 5.22 = -3.22 (not equivalent to 0.0841)

Therefore, none of the expressions given is equivalent to (2 - ³√24)².

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The remainder when 7x⁴ - 3x is divided by x - 1 is 4.

knοwn as indeterminates) and cοefficients, which are cοmbined using the οperatiοns οf additiοn, subtractiοn, and multiplicatiοn. A pοlynοmial can have οne οr mοre variables, but each term in the pοlynοmial must have nοn-negative integer expοnents οn the variables. The degree οf a pοlynοmial is the highest pοwer οf its variables with a nοn-zerο cοefficient.

Fοr example, the pοlynοmial 3x² - 2x + 5 has a degree οf 2, with the term 3x² being the highest degree term. The cοefficient οf the term 3x^2 is 3, and the cοefficient οf the term -2x is -2.

Pοlynοmials are used in a variety οf mathematical applicatiοns, including algebra, calculus, and geοmetry. They are used tο represent mathematical functiοns, tο apprοximate cοmplex curves, and tο sοlve equatiοns. Sοme cοmmοn οperatiοns οn pοlynοmials include additiοn, subtractiοn, multiplicatiοn, divisiοn, and factοring.

Tο find the remainder when the pοlynοmial 7x⁴ - 3x is divided by x - 1, we can use pοlynοmial lοng divisiοn οr synthetic divisiοn.

7x³ + 7x² + 7x + 4

x - 1 | 7x⁴ + 0x³ - 3x² + 0x + 0

     - (7x⁴ - 7x³)

           7x³ - 3x²

           - (7x³ - 7x²)

                 4x² + 0x

                 - (4x² - 4x)

                        4x

                        - (4x - 4)

                             4

Therefore, the remainder when 7x⁴ - 3x is divided by x - 1 is 4.

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The  interval notation  of x^(2 )-2x-35>0  is (-∞,-5)∪(7,∞).

To solve the quadratic inequality x^(2)-2x-35>0, we first need to find the roots of the quadratic equation x^(2)-2x-35=0. We can do this by factoring the equation:

(x-7)(x+5)=0

The roots of the equation are x=7 and x=-5. Now, we can use these roots to determine the intervals where the inequality is true. We can do this by testing values in each interval:

- For x<-5, let's test x=-6: (-6)^(2)-2(-6)-35=1>0, so the inequality is true in this interval.
- For -57, let's test x=8: (8)^(2)-2(8)-35=29>0, so the inequality is true in this interval.

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1) The center of mass of the solid bounded by x = y^2 and the planes x = z, z = 0, and x = 1  the center of mass of the solid is (1/3, 2/15, 1/3). 2) The total charge on the disk is 4/3 coulombs. 3) The center of mass of the triangular region is (2/3, 2/3).


Center of Mass = (∫xyzρdV)/(∫ρdV).

Here, V is the volume of the solid. Since the density is constant, we can pull it out of the integral:

Center of Mass = k*(∫xyzdV)/(∫dV).

We can now use the volume formula for the solid which is V = ∫xyzdxdyz. Plugging this in the above formula, we get:

Center of Mass = k*[(∫x∫ydxdyz)/(∫dxdyz)]

Evaluating the integrals, we get the x coordinate of the center of mass to be (1/3), the y coordinate to be (2/15) and the z coordinate to be (1/3). Thus, the center of mass of the solid is (1/3, 2/15, 1/3).

2. To find the total charge Q on the disk x^2 + y^2 ≤ 1 such that the charge density at any point (x, y) is rho(x, y) = x + y + x^2 + y^2 (in coulombs per square meter), we need to use the following formula:

Q = ∫∫rho(x, y)dxdy

Evaluating the integral, we get Q = (1/3) + (1/3) + (1/3) + (1/3) = 4/3. Thus, the total charge on the disk is 4/3 coulombs.

3. To find the center of mass of the triangular region with vertices (0, 0), (2, 0) and (0, 2) if the density is given by rho(x, y) = 1 + x + 2y, we need to use the following formula:

Center of Mass = (∫xyρdA)/(∫ρdA).

Here, A is the area of the triangle. Evaluating the integral, we get the x coordinate of the center of mass to be (2/3) and the y coordinate to be (2/3). Thus, the center of mass of the triangular region is (2/3, 2/3).

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Answer:

We are not given the ratio of cordial to water used in the mixture, so we can assume that the entire bottle of cordial is mixed with water to make the drink.

Since the bottle of cordial holds 800 ml of cordial, the total volume of the mixture would be 800 ml + volume of water added. Let's call the volume of water added x.

Therefore, the total volume of the drink would be 800 ml + x.

We are asked to find the minimum capacity of the container in liters, so we need to convert the total volume of the drink from milliliters to liters:

800 ml + x = (800 + x)/1000 liters

Now we can set up an inequality to find the minimum value of x that would make the total volume of the drink at least 1 liter:

800 ml + x ≥ 1000 ml

Simplifying this inequality, we get:

x ≥ 200 ml

Therefore, the minimum volume of water that needs to be added to the cordial to make a drink with a total volume of at least 1 liter is 200 ml.

So the minimum capacity of the container would be:

800 ml + 200 ml = 1000 ml = 1 liter

Therefore, the minimum capacity of the container in liters would be 1 liter.

Step-by-step explanation:

Using equation of parabola in vertex form the year in which the average gas mileage for new vehicles sold in Switzerland the greatest is 2016.

The equation of a parabola with vertex (h, k) is given by

y = a(x - h)² + k

Now a researcher found that for the years 2013 to 2019, the equation, y = -0.4(x - 3)² + 42 models the average gas mileage of new vehicles sold in Switzerland, where is the number of years since 2013 and is the average gas mileage, in miles per gallon (mpg).

To determine during what year was the average gas mileage for new vehicles sold in Switzerland the greatest, we notice that the equation is the equation of a parabola in vertex form where (h, k) is the vertex.

Comparing y = a(x - h)² + k with y = -0.4(x - 3)² + 42 we have that

a = -0.4, h = 3 and k = 42

So, the vertex is at (h, k) = (3, 42)

Since a = -0.4 < 0, (3,42) is a maximum point

So, y is maximum when x = 3

Since this is 3 years after 2013 which is 2013 + 3 = 2016.

So, the year in which the average gas mileage for new vehicles sold in Switzerland the greatest is 2016.

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The quotient is 4x+5 and the remainder is -15x.

The quotient and remainder when dividing (12x^(3)+15x^(2)+21x)/(3x^(2)+4) can be found using polynomial long division.

First, divide the leading term of the numerator by the leading term of the denominator: (12x^(3))/(3x^(2)) = 4x. This is the first term of the quotient.

Next, multiply the first term of the quotient by the denominator and subtract the result from the numerator: (12x^(3)+15x^(2)+21x) - (4x)(3x^(2)+4) = 15x^(2)+5x.

Now, repeat the process with the new numerator: (15x^(2))/(3x^(2)) = 5. This is the second term of the quotient.

Again, multiply the second term of the quotient by the denominator and subtract the result from the new numerator: (15x^(2)+5x) - (5)(3x^(2)+4) = -15x.

Since the degree of the new numerator is less than the degree of the denominator, the division is complete and the new numerator is the remainder.

Therefore, the quotient is 4x+5 and the remainder is -15x.

In conclusion, the quotient and remainder when dividing (12x^(3)+15x^(2)+21x)/(3x^(2)+4) are 4x+5 and -15x, respectively.

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Option 4: 2 feet, 3 feet, and 5 feet – constitutes a right angle since 2² + 3² = 4 + 9 = 13 and 13 = 5², making it.

A triangle is a geometry with three vertices and three sides. The internal angle of the triangle, which really is 180 degrees, is built. The inner triangle angles are implied to sum to 180 degrees. It has the fewest sides of any polygon.

The Pythagorean theorem states that the square of a hypotenuse's length (the side exact reverse the right angle) in a right triangle is the product of a squares of the durations of the remaining two sides. Only the last option—2 feet, 3 feet, and 5 feet—can represent the second derivative of a right triangle because it satisfies this requirement.

Let's check each option:

Option 1: 14 feet, 14 feet, 14 feet - As all three are equal, this doesn't qualify as a right triangle and the Pythagoras theorem cannot be met.

Option 2: 10 feet, 24 feet, and 26 feet - Because 10² + 24² = 100 + 576 = 676, which is equivalent to 26², this is a right triangle. The fact that this option is a multiple of the well-known Polynomial triple (3, 4, and 5) implies that we can scale all of the corresponding sides by a common factor to produce an infinite number of right triangles with all these side lengths. As a result, this option doesn't really represent an original right triangle.

Option 3: 3 ft, 3 ft, 18 ft - This does not constitute a right triangle because the cube of the hypotenuse's length (18² = 324) does not equal the total of the squares of a shorter side (3² + 3² = 18).

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Answer:

2 feet, 3 feet, and 5 feet

Step-by-step explanation:

got it right on my test

The estimate of the probability that he will hit at least 6 times in his next 10 attempts is given as follows:

80%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

An amateur golfer hits the ball 48% of the time he attempts, hence we round the probability to 50%, and have that the numbers are given as follows:

From the table, we have 20 sets of 10 attempts, and in 16 of them he hit at least 6 attempts, hence the probability is given as follows:

16/20 = 0.8 = 80%.

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