The area of the figure is square units.

3 units, 8 units, 3 units, 9 units, 3 units, 21 units

Answer:

math = 227.98

programming = 42.55

Step-by-step explanation:

We have
3m + 4p = 854.14 -eq(1)

8m + 1p = 1866.39 -eq(2)

rq(2) x 4: 32m + 4p = 7465.56 -eq(3)

eq(3)-eq(1):

32m + 4p = 7465.56

- ( 3m + 4p = 854.14)

--------------------------------

29m = 6611.42

--------------------------------

⇒ m = 6611.42/29

m = 227.98

sub in eq(1)

3(227.98) + 4p = 854.14

4p = 854.14 - 683.94

4p = 170.2

p = 170.2/4

p = 42.55

The indicated operation is the composition of functions. To perform this operation, we substitute the expression for g(x) into f(x). The composition of f(g(x)) is given by f(g(x)) = (4x - 1)^2 + 2.

To compute f(g(x)), we first evaluate g(x) by substituting x into the expression for g(x): g(x) = 4x - 1. Next, we substitute this result into f(x): f(g(x)) = f(4x - 1).

Now, let's expand and simplify f(g(x)):

f(g(x)) = (4x - 1)^2 + 2

        = (4x - 1)(4x - 1) + 2

        = 16x^2 - 8x + 1 + 2

        = 16x^2 - 8x + 3.

The domain of f(g(x)) is the same as the domain of g(x) since the composition involves g(x). In this case, g(x) is defined for all real numbers. Therefore, the domain of f(g(x)) is also all real numbers.

In summary, the composition of f(g(x)) is given by f(g(x)) = 16x^2 - 8x + 3, and the domain of f(g(x)) is all real numbers.

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The estimated standard deviation of the bear weights, based on the given graph and using the 69-95-99.7% rule, is approximately 63.

The 69-95-99.7% (empirical) rule, also known as the 3-sigma rule, is a rule of thumb that applies to data that follows a normal distribution. According to this rule:

- Approximately 68% of the data falls within one standard deviation of the mean.

- Approximately 95% of the data falls within two standard deviations of the mean.

- Approximately 99.7% of the data falls within three standard deviations of the mean.

In the given graph, if we assume the bear weights follow a normal distribution, we can estimate the standard deviation using the 69-95-99.7% rule.

Based on the graph, we know that the midpoint of the distribution (mean) is 200. Assuming the graph is symmetric, we can estimate one standard deviation as half the distance between the mean (200) and either end (74 or 326).

To calculate this, we subtract the mean from one of the endpoints and divide by 2:

Standard Deviation ≈ (326 - 200) / 2 ≈ 126 / 2 ≈ 63

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Answer:

[tex]x = \frac{3\pi}{64} +\frac{9}{16}[/tex]

Step-by-step explanation:

We have

∠D = 45° = 45* π/180 radians = π/4 radians - eq(1)

big arc + small arc = 2π

small arc = 16x - 9

⇒ big arc = 2π - small arc

big arc = 2π - 16x + 9

[tex]\angle D = \frac{big \;arc - small \;arc}{2}[/tex]

[tex]\angle D = \frac{2\pi - 16x + 9 - 16x +9}{2}\\\\= \angle D = \frac{2\pi - 32x + 19 }{2}\\\\\angle D = \pi - 16x + 9[/tex]

Equating with eq(1)

π - 16x + 9 = π/4

⇒ 16 x = π - (π/4) +9

⇒ 16 x = (3π/4) +9

⇒ [tex]x = \frac{1}{16} (\frac{3\pi}{4} +9)[/tex]

[tex]x = \frac{3\pi}{64} +\frac{9}{16}[/tex]

Answer:

violence figer in the past two years

Answer:

[tex]x = 5\sqrt3[/tex]

Step-by-step explanation:

We can solve for the side length x in this 30-60-90 triangle by using the ratio of side lengths for that specific type of right triangle:

In this triangle, we can identify the smallest side (corresponding to 1 in the ratio) as 5. This means we can solve for x by multiplying 5 by [tex]\sqrt3[/tex]. Thus:

[tex]\boxed{x = 5\sqrt3}[/tex]

Answer:

[tex]_{13}C_5=1287[/tex]

Step-by-step explanation:

[tex]\displaystyle _nC_r=\frac{n!}{r!(n-r)!}\\\\_{13}C_5=\frac{13!}{5!(13-5)!}\\\\_{13}C_5=\frac{13!}{5!\cdot8!}\\\\_{13}C_5=\frac{13*12*11*10*9}{5*4*3*2*1}\\\\_{13}C_5=\frac{154440}{120}\\\\_{13}C_5=1287[/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5!(13 - 5)!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5!(8)!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5! \times 8!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8!}{5! \times 8!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 \times \cancel{8!}}{5! \times \cancel{8!}} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 }{5 \times 4 \times 3 \times 2 \times 1} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{156 \times 110 \times 9 }{20 \times 6 \times 1} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{156 \times 990 }{20 \times 6 } \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{154440 }{120 } \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = 1287 \\ [/tex]

The regression equation that correctly models the data is: y = 5.71x + 27.6.

The correct answer to the given question is option D.

Regression equations are mathematical models that relate two or more variables to find the relationship between them. One variable, denoted as y, is considered the dependent variable. The other variable, denoted as x, is considered the independent variable.

In this case, the independent variable is the size of the outdoor deck, while the dependent variable is the estimated cost to construct it.

There are different types of regression equations. The one that fits this scenario is the linear regression equation, which has the form y = mx + b, where m is the slope of the line and b is the y-intercept.

The slope represents the change in y for each unit change in x, while the y-intercept represents the value of y when x is zero. To find the regression equation that correctly models the data, we need to calculate the slope and the y-intercept using the given values.

We can use the following formulas:

Slope:  m = [(n∑xy) - (∑x)(∑y)] / [(n∑x2) - (∑x)2]

Y-intercept: b = (∑y - m∑x) / n Where n is the number of data points, which is 6 in this case.

Using the given values, we get: Slope: m = [(6)(2,010,500) - (1,193)(6,950)] / [(6)(346,337) - (1,193)2] = 5.71

Y-intercept: b = (6,950 - (5.71)(1,193)) / 6 = 27.6

Therefore, the regression equation that correctly models the data is: y = 5.71x + 27.6

The answer is option D.

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The calculated equation of the line is y = -2x + 3

From the question, we have the following parameters that can be used in our computation:

The graph

Where, we have

(1, 1) and (0, 3)

The equation of the line is calculated as

y = mx + c

Where

c = y when x = 0

So, we have

y = mx + 3

Using the points, we have

m + 3 = 1

So, we have

m = -2

This means that

y = -2x + 3

Hence, the equation of the line is y = -2x + 3

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Answer:182.12 meters of fencing required.

Step-by-step explanation:

The calculated value of x in the triangle is 40°

From the question, we have the following parameters that can be used in our computation:

The figure (see attachment)

Using the theorem of linear pair, we have

∠DBA + ∠ABC = 180°

Using the given values, we have

∠100° + ∠ABC = 180°

Collect the like terms

∠ABC = 180° - 100°

Evaluate

∠ABC = 80°

The sum of angles of a triangle is 180° .

So, in triangle ABC

∠A + ∠B + ∠C = 180°

Using the given values, we have

x + 80 + 60 = 180°

Evaluate the sum

x + 140 = 180°

Collect the like terms

x = 180° - 140°

Evaluate

x = 40°

Hence, the value of x is 40°

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The correct statement is OB. The average rate of change of f is more than the average rate of change of g.

To determine the average rate of change (slope) of the functions f and g, we can use the formula:

Average Rate of Change = (f(x2) - f(x1)) / (x2 - x1)

For function f, using the given table, we can calculate the average rate of change between the points (0, 0) and (2, -2):

Average Rate of Change (f) = (-2 - 0) / (2 - 0) = -2 / 2 = -1

For function g, using the given points (-3, 5) and (0, 14), we can calculate the average rate of change:

Average Rate of Change (g) = (14 - 5) / (0 - (-3)) = 9 / 3 = 3

Comparing the average rates of change, we find that the average rate of change of f is -1, while the average rate of change of g is 3.

Therefore, the correct statement is:

OB. The average rate of change of f is more than the average rate of change of g.

The average rate of change of f is greater than the average rate of change of g, indicating that the function f is increasing at a faster rate than function g over the given interval.

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Answer:

6b + 5n = 101

Step-by-step explanation:

The correct system of equations that can be used to find b, the number of bracelets Ily sold, and n, the number of necklaces she sold is:

b + n = 18

6b + 5n = 101

In this system, the first equation represents the total number of items sold, which is 18. Since b represents the number of bracelets and n represents the number of necklaces, the equation b + n = 18 reflects that the total number of bracelets and necklaces sold should add up to 18.

The second equation represents the total amount of money Ily earned from selling bracelets and necklaces. Since bracelets were sold for $6 each and necklaces for $5 each, the equation 6b + 5n = 101 represents the total amount of money earned, which is $101.

Therefore, the correct system of equations is:

b + n = 18

6b + 5n = 101

The ordered pairs which are in the solution set of the system of linear inequalities are (5, –2), (3, 1), (4, 2).

The correct answer to the given question is option C.

The system of linear inequalities is:y > -1/2 x y < 1/2 x + 1

On a coordinate plane, two straight lines are shown.

The first solid line has a negative slope and goes through (0, 0) and (4, -2).

Everything above the line is shaded.

The second dashed line has a positive slope and goes through (-2, 0) and (2, 2).

Everything below the line is shaded.

We will check which of the given ordered pairs lie in the solution set of this system of linear inequalities. 1. (5, -2) Putting x = 5 and y = -2, we get:

y > -1/2 x   ⇒   -2 > -1/2 (5)   ⇒   -2 > -2.5  which is false.

xy < 1/2 x + 1   ⇒   (5)(-2) < 1/2 (5) + 1   ⇒   -10 < 3.5  which is false.

Therefore, the ordered pair (5, -2) is not in the solution set of the system of linear inequalities. 2. (3, 1) Putting x = 3 and y = 1, we get:

y > -1/2 x   ⇒   1 > -1/2 (3)   ⇒   1 > -1.5  which is true.

xy < 1/2 x + 1   ⇒   (3)(1) < 1/2 (3) + 1   ⇒   3 < 2.5  which is false.

Therefore, the ordered pair (3, 1) is not in the solution set of the system of linear inequalities. 3. (-4, 2) Putting x = -4 and y = 2, we get:y > -1/2 x   ⇒   2 > -1/2 (-4)   ⇒   2 > 2  which is false. xy < 1/2 x + 1   ⇒   (-4)(2) < 1/2 (-4) + 1   ⇒   -8 < -0.5  which is true.

Therefore, the ordered pair (-4, 2) is in the solution set of the system of linear inequalities. Hence, the answer is (5, –2), (3, 1), (4, 2).

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Answer:

D. 48

Step-by-step explanation:

When a tangent and a secant, two secants, or two tangents intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

56 = 1/2(160 - ?)

112 = 160 - ?

? = 160 - 112 = 48

Answer:

A: $51.00

B: $54.57

Step-by-step explanation:

Let amount = $68.00

Part A:

Since the coupon is for 25%, the family pays 75% of $68.00

75% of $68.00 = 0.75 × $68.00 = $51.00

The new price is $51.00

Part B:

The tax is 7% of $51.00

7% of $51.00 = $3.57

The total price is the sum of $51.00 and the amount of tax, $3.57

Total price = $51.00 + $3.57 = $54.57

The center of the ellipse defined by the equation 9x^2 + 4y^2 + 18z - 23 = 0 is (0, 0, 0).

9x^2 + 4y^2 + 18z - 23 = 0 must be rearranged to its standard form in order to determine the location of the ellipse's centre. The ellipse equation has the following standard form:

(x - h)^2/a^2 + (y - k)^2/b^2 + (z - l)^2/c^2 = 1,

where (h, k, l) represents the center of the ellipse, and a, b, and c are the semi-major, semi-minor, and semi-vertical axes, respectively.

Let's rearrange the given equation to match the standard form:

9x^2 + 4y^2 + 18z - 23 = 0

Dividing by 23 to simplify the equation:

(9x^2)/23 + (4y^2)/23 + 18z/23 - 1 = 0

Now, we can rewrite the equation as:

(9x^2)/23 + (4y^2)/23 + 18z/23 = 1

Comparing this with the standard form, we can identify the values of a, b, and c:

a^2 = 23/9

b^2 = 23/4

c^2 = 23/18

Taking the square roots of these values:

a ≈ √(23/9) ≈ 1.53

b ≈ √(23/4) ≈ 1.92

c ≈ √(23/18) ≈ 1.23

Therefore, the semi-major axis (a) is approximately 1.53, the semi-minor axis (b) is approximately 1.92, and the semi-vertical axis (c) is approximately 1.23.

The center of the ellipse is represented by the values (h, k, l). Since the equation does not involve any shifts or translations, the center is located at the origin (0, 0, 0).

As a result, (0, 0, 0) is the centre of the ellipse formed by the equation (9x2 + 4y2 + 18z - 23 = 0).

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Answer:

Center: (-3, 1)

Step-by-step explanation:

The center of an ellipse is always the point that is equidistant from the two foci and the two vertices.

In order to find the center of the ellipse, we can complete the square in both the x and y terms.

First, we move the constant term to the right side of the equation:

[tex]9x^2+18x + 4y^2-8y = 23[/tex]

The coefficient of x is 18, so half of it is 9, and squaring that gives us 81. Adding 81 to both sides of the equation gives us:

[tex]9x^2+18x + 81 = 23 + 81[/tex]

which combines the terms in x into a squared term:

[tex](9x+9)^2 = 104[/tex]

Similarly:

[tex]4y^2-8y + 4 = 23 + 4[/tex]

which combines the terms in y into a squared term:

[tex](2y-2)^2 = 27[/tex]

Now that we have completed the square in both x and y, we can write the equation in standard form for an ellipse:

[tex]\frac{(x+3)^2}{11^2} + \frac{(y-1)^2}{3^2} = 1[/tex]

The standard form for an ellipse is:

[tex]\boxed{\bold{\frac{(x-h)^2}{a^2} +\frac{ (y-k)^2}{b^2} =1 }}[/tex]

where (h, k) is the center of the ellipse, a is the radius in the x-direction, and b is the radius in the y-direction.

In the equation for our ellipse, (h, k) = (-3, 1).

So the center of the ellipse is at the point (-3, 1).

In the nearest tenth, this is (-3.0, 1.0).

Answer:

The one at the bottom right above the next button

Step-by-step explanation:

Answer:

(a)  $556 billion

(b)  $581 billion

(c)  $693 billion

Step-by-step explanation:

The given function is:

[tex]\boxed{A(x)=314e^{0.044x}}[/tex]

where A(x) is the assets (in billions of dollars) for a financial firm .

If x = 7 corresponds to the year 2007 then:

Therefore, to find the assets for each of the given years, substitute the corresponding value of x into the function.

[tex]\begin{aligned}A(13)&=314e^{0.044 \cdot 13}\\&=314e^{0.572}\\&=314(1.77180712...)\\&=556.34743707...\\&=556\; \sf (nearest\;billion)\end{aligned}[/tex]

[tex]\begin{aligned}A(14)&=314e^{0.044 \cdot 14}\\&=314e^{0.616}\\&=314(1.851507181...)\\&=581.3732549...\\&=581\; \sf (nearest\;billion)\end{aligned}[/tex]

[tex]\begin{aligned}A(18)&=314e^{0.044 \cdot 18}\\&=314e^{0.792}\\&=314(2.20780762...\\&=693.2515954...\\&=693\; \sf (nearest\;billion)\end{aligned}[/tex]

The sample variance (s²) is approximately 29.5 and the sample standard deviation (s) is approximately 5.4.

To find the sample variance and standard deviation,

Calculate the mean (average) of the given data set.

21 + 12 + 6 + 7 + 10 = 56

Mean = 56 / 5 = 11.2

Square the result of subtracting the mean from each data point.

(21 - 11.2)² = 96.04

(12 - 11.2)² = 0.64

(6 - 11.2)² = 27.04

(7 - 11.2)² = 17.64

(10 - 11.2)² = 1.44

Calculate the sum of the squared differences

96.04 + 0.64 + 27.04 + 17.64 + 1.44 = 142.8

Divide the sum by (n-1), where n is the number of data points (in this case, 5).

142.8 / (5-1) = 35.7

The result is the sample variance (s²).

Take the square root of the sample variance to determine the sample standard deviation (s).

s = √35.7 ≈ 5.4

Therefore, the sample variance is approximately 29.5 (rounded to one decimal place) and the sample standard deviation is approximately 5.4 (rounded to one decimal place).

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Answer:

  A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.

Step-by-step explanation:

You want to know the error that adding -4+4 on the left and 1+1 on the right represents in the solution of the equation ...

The correct solution procedure would be to eliminate the parentheses using the distributive property as a first step:

  4x -6x +8 +4 = -x +3x +3 +1

Comparing this to Maricella's first step, we see that she ignored the step of using the distributive property to multiply the constants inside parentheses by the factor outside. The appropriate description of Maricella's mistake is ...

A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.

__

Additional comment

Adding like terms would give ...

  -2x +12 = 2x +4

  8 = 4x . . . . . . . . . . . add 2x-4 to both sides

  2 = x . . . . . . . . . . divide by 4

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The length of the shortest path ABCD is 7 units.

To find the length of the shortest path ABCD, we need to determine the coordinates of points B and C and then calculate the distances between these points.

Given that B has a y-coordinate of 2, it lies on a horizontal line. Therefore, the y-coordinate of B is 2, and the x-coordinate is the same as the x-coordinate of point A, which is 7. So, B is the point (7, 2).

Similarly, C lies on a vertical line, and its x-coordinate is the same as the x-coordinate of point D, which is -5. So, C is the point (-5, 0).

Now, we can calculate the distances between the points. The distance between A and B can be found using the distance formula:

AB = √[tex]((x2 - x1)^2 + (y2 - y1)^2[/tex])

Substituting the coordinates of A and B, we have:

AB = √[tex]((7 - 7)^2 + (2 - 4)^2) = √(0^2 + (-2)^2[/tex]) = √4 = 2

The distance between B and C is simply the difference in their y-coordinates:

BC = |y2 - y1| = |2 - 0| = 2

Finally, the distance between C and D can be calculated using the distance formula:

CD = √[tex]((-5 - (-5))^2 + (0 - (-3))^2)[/tex] = √[tex](0^2 + 3^2)[/tex] = √9 = 3

Therefore, the length of the shortest path ABCD is the sum of the distances AB, BC, and CD:

Shortest path ABCD = AB + BC + CD = 2 + 2 + 3 = 7

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Answer:

Step-by-step explanation:

To find the density of the wooden board, we need to divide the mass of the board by its volume. The volume of a rectangular prism is calculated by multiplying its length, width, and height.

Given:

Length (l) = 0.3 m

Width (w) = 2.1 m

Height (h) = 0.1 m

Mass (m) = 0.17 kg

Volume (V) = l × w × h

V = 0.3 m × 2.1 m × 0.1 m

V = 0.063 m³

Density (ρ) = mass / volume

ρ = 0.17 kg / 0.063 m³

ρ ≈ 2.7 kg/m³

The density of the wooden board is approximately 2.7 kg/m³.

Answer:

Angle C must measure 40 degrees.

Step-by-step explanation:

All angles in a triangle add up to 180 degrees

(90-50)=40 degrees

We can check our answer by adding all the angles up

90+50+40=180

Angle C must be 40 degrees

The function rule that describes the given translation is T-9, 1(x, y).

The first value in the function rule represents the horizontal translation, while the second value represents the vertical translation. In this case, the square is translated 1 unit to the right, indicating a positive horizontal translation.

Additionally, the square is translated 9 units down, indicating a negative vertical translation. Therefore, the correct function rule is T-9, 1(x, y).

In the coordinate plane, the x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position. When we apply the function rule T-9, 1 to the coordinates of the square, we subtract 9 from the y-coordinate and add 1 to the x-coordinate.

This results in the square being moved 9 units down and 1 unit to the right from its original position.

The negative sign in front of the 9 indicates a downward movement, and the positive sign in front of the 1 indicates a rightward movement. Hence, the translation is accurately described by the function rule T-9, 1(x, y).

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Answer:

C

Step-by-step explanation:

Answer:

Para resolver este problema, podemos plantear un sistema de ecuaciones. Si definimos "p" como el número de velas pequeñas y "g" como el número de velas grandes, podemos expresar la información del problema de la siguiente manera:

p + g = 20 (la tienda vendió un total de 20 velas) 10.98p + 27.98g = 355.60 (el ingreso total por la venta de velas fue de $355.60)

Podemos resolver este sistema de ecuaciones utilizando el método de sustitución. Despejando "p" de la primera ecuación, obtenemos:

p = 20 - g

Luego, sustituimos esta expresión de "p" en la segunda ecuación:

10.98(20 - g) + 27.98g = 355.60

220.20 - 10.98g + 27.98g = 355.60

17.00g = 135.40

g = 8

Por lo tanto, la tienda vendió 8 velas grandes.

Step-by-step explanation:

Step-by-step explanation:

Part A: The graph of the system of inequalities consists of two lines and a shaded region.

The line 2x + 3y = 15 is a solid line (because of the "less than" symbol in the inequality) and is graphed using a straight line connecting two points. For example, when x = 0, y = 5, and when x = 7.5, y = 0.

The line x + y = 3 is a solid line (because of the "greater than or equal to" symbol in the inequality) and is graphed using a straight line connecting two points. For example, when x = 0, y = 3, and when x = 3, y = 0.

The shaded region represents the solution set. It is the area below the line 2x + 3y = 15 and above or on the line x + y = 3. This shaded region satisfies both inequalities simultaneously.

Part B: To determine if the point (5, 1) is included in the solution area, we substitute x = 5 and y = 1 into both inequalities:

2x + 3y < 15:

2(5) + 3(1) < 15

10 + 3 < 15

13 < 15

Since 13 is less than 15, the point (5, 1) satisfies the first inequality.

x + y ≥ 3:

5 + 1 ≥ 3

6 ≥ 3

Since 6 is greater than or equal to 3, the point (5, 1) satisfies the second inequality.

Since the point (5, 1) satisfies both inequalities, it is included in the solution area for the system.

Part C: Let's choose the point (2, 2) as another example from the solution set.

Interpretation in real-world context:

When we have x = 2 and y = 2, it means Michael decides to buy 2 cupcakes and 2 pieces of fudge. This combination of sweets satisfies the conditions set in the inequalities, ensuring that he can feed at least three siblings.

The point (2, 2) represents a valid solution in which Michael spends a total of $10 (2 cupcakes * $2/cupcake + 2 fudges * $3/fudge = $4 + $6 = $10). With this choice, he can afford to buy enough treats to feed his three siblings while staying within his budget of $15.

The side length that prove a right angle triangle is √2, √3 and √5.

A right angle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, a right angle triangle can be proved by using the Pythagoras's theorem as follows:

Hence,

c² = a² + b²

where

Therefore,

(√2)² + (√3)² = (√5)²

Hence, the right angle triangle is the triangle with sides √2, √3 and √5.

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Answer:

Isaac Newton (late 17th century) - Newton made significant contributions to calculus and mathematical physics. His book "Philosophiæ Naturalis Principia Mathematica" laid the groundwork for classical mechanics.