Solve the system of linear equations by substitution.

y=4x
x+y=-10

Answer:

7a) To show that r^2 = R^2 - h^2, one can use the Pythagorean theorem and the fact that the distance from the center of the sphere to the surface of the cylinder is h/2.

7b) To find the volume of the cylinder, one can use the formula for the volume of a cylinder, V = πr^2h, where r is the radius and h is the height.

7c) To find the maximum volume of the cylinder, one can differentiate the expression for the volume with respect to h, set the derivative equal to zero, and solve for h.

7d) The ratio of the volume of the sphere to the maximum volume of the cylinder can be found by dividing the volume of the sphere by the maximum volume of the cylinder.

To find the ratio of radius to height for the desired can, one can set up an expression for the surface area of the can and differentiate it with respect to the radius and height. Setting the partial derivatives equal to zero, one can find the critical points, and evaluate which one is the minimum.

10a) To find the area of the rectangle, one can use the formula for the area of a rectangle, A = xy, where x is the length and y is the width.

10b) To find the maximum area of the rectangle, one can differentiate the expression for the area with respect to x and y, set the partial derivatives equal to zero, and solve for x and y.

Step-by-step explanation:

Step 1: Count the total number of students

The total number of students in the statistics class is 8.

Step 2: Count the number of women

There are 4 women in the statistics class.

Step 3: Calculate the frequency, proportion, and percentage of women

Frequency: The frequency of women in the statistics class is 4.

Proportion: The proportion of women in the statistics class is 4/8, or 0.5.

Percentage: The percentage of women in the statistics class is 50%.

The information in the question is not significant to calculate the frequency, proportion and percentage of women.

Statistical models for calculating gender through frequency involve counting the number of individuals in a population that identify as each gender. This can be done by collecting data on gender from surveys, census data, or other sources of information. The data collected is then used to calculate the frequency, proportion, and percentage of each gender in the population. This information can be used to better understand gender dynamics in a population and inform policies and practices that promote gender equity.

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On solving the linear equations - 9x - 3y = -21 and 9x + 9y = -9 the value of x is x = 4 and y is y = -5.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

The linear equations are -

- 9x - 3y = -21 .... (1)

9x + 9y = -9 .... (2)

Add the equations (1) and (2) -

- 9x - 3y + 9x + 9y = -21 - 9

6y = -30

y = -5

Substitute the value of y in equation (1) -

- 9x - 3(-5) = -21

- 9x + 15 = -21

- 9x = -36

-x = -4

x = 4

Therefore, the values of x and y is 4 and -5.

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Answer:

Step-by-step explanation:

   Ted's balcony

       |\

       | \

       |  \ <- 40 degrees

       |   \

       |    \  <-- line of sight to the dog

       |     \

       |      \

       |       \

       |        \

       |         \

       |__________\

          Ted's

          eye level

Let d be the distance the dog needs to walk to get to the donut. We can use trigonometry to solve for d.

Since the angle of depression from Ted's line of sight to the dog is 40 degrees, the angle between the ground and Ted's line of sight is 90 - 40 = 50 degrees.

We can use the tangent function to find d:

tan(50) = opposite / adjacent

The opposite side is the height of Ted's eye level above the ground, which is 16 feet. The adjacent side is d, the distance the dog needs to walk to get to the donut.

tan(50) = 16 / d

d = 16 / tan(50)

d ≈ 12.6 feet

Answer:

183.25

Step-by-step explanation:

5.56 × 32.958

= 183.24648

Estimated to 2 Decimal Places:

183.25

Answer:

6

Step-by-step explanation:

As you can see from the graph, it intercepts the y-axis at -4 and x-axis at 6

Answer:

[tex]\textsf{A)} \quad \dfrac{m\overline{AC}}{m\overline{DF}}=3[/tex]

Step-by-step explanation:

In similar triangles:

Given ΔABC ~ ΔDEF with a scale of 3 : 1 then:

    [tex]\bullet \quad \overline{AB} = 3\overline{DE}[/tex]

    [tex]\bullet \quad \overline{BC} = 3\overline{EF}[/tex]

    [tex]\bullet \quad \overline{AC} = 3\overline{DF}[/tex]

Therefore:

[tex]\dfrac{m\overline{AC}}{m\overline{DF}}=3[/tex]

There are 172 boys in the introductory geology course.

The ratio of boys to girls is 4:5. We can know that the number of boys in the class is four multiplied by the common value that the ratio is based on.

We can call this usual value x and say that the number of boys in the class is 4x. Applying the same logic to the girls, we can say that there are 5x girls in the class. Hence, a total of 9x children. We divide the total value by 9 to get x, which equals 43.

We can then calculate the number of boys by calculating 4x, which equals 172.

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To find a percentage of a whole, you first need to know exactly what you're measuring. For instance, if 38 people in your community have library cards, and you wish to work out the percentage of library card holders in the population, you'll first need to know the total size of your community.

A percentage of a total is equal to the subset of a total divided by the total, then multiplied by 100. If your community has 230 people in it, you can calculate the percentage of people with a library card. First, you divide 38 by 230, then multiply that total by 100.

38/230 = .165 .165 x 100 = 16.5

Therefore 16.5 percent of the people in your community have a library card.

A fraction has two parts, a numerator on the top and a denominator on the bottom. Any percentage can be expressed as a fraction with the percentage value as the numerator and 100 as the denominator. So 80 percent is also 80/100. You can use this to convert a fraction into a percentage as well.

For example, to turn the fraction 4/25 into a percent, you'll need a fraction with 100 as the denominator. So you'll need to multiply the denominator by 4 since this will produce 100. You then need to also multiply the numerator by the same amount.

4/25 = (4 x 4)/(25 x 4) = 16/100

You now have a fraction of 100, so you can convert it to a percentage quite easily: 16/100 is equal to 16 percent. Here's another example:

3/5 = (3 x 20)/(5 x 20) = 60/100 = 60%

If you have access to a calculator, you could also divide the numerator by the denominator and turn the fraction into a decimal. Decimals are simple to convert into percent notation.

Decimals are even easier to convert into percentages. To convert a decimal into a percentage, multiply the decimal by 100 and add the percent sign. To multiply a decimal by 100, move the decimal point two digits to the right:

0.8= 0.8 x 100 = 80% 0.53: 0.53 x 100 = 53% 0.173: 0.173 x 100 = 17.3%

You can also use this method to convert decimal numbers that are larger than 1. Say you know there are 1.34 times as many books in your library this year than last year, and you want to know what percentage the number of books you have this year is of the books you had last year. As before, you would multiply 1.34 by 100 and add the percent sign:

1.34 x 100 = 134%

Your number exceeds 100 percent because you're comparing this year's book total to the total books from last year, a number represented by 100 percent. Since there are more books this year, that percentage is larger than 100.

You can convert percentages to decimals using the opposite operation. Instead of multiplying the decimal by 100, divide the percent by 100. If 56 percent of people eat roast beef sandwiches you could also say that for every 100 people, 56 eat roast beef. Divide 56 by 100. Because you're dividing by 100, you'll move the decimal point two places to the left:

56/100 = 0.56

Other conversion examples are: 5%: 5/100 = 0.05, 79%: 79/100 = 0.79, 295%: 295/100 = 2.95

The volume of the cone = 0.262 feet³.

The word "cone" is derived from the Greek word "konos," which denotes a peak or a wedge. Apex and base both refer to the pointy end, which is referred to as the apex. A cone is a three-dimensional geometric structure with a smooth transition from a flat, generally circular base to the apex or vertex, a point that creates an axis to the base's center.

Given that,

base of the cone = 1 foot

thus, radius (r) = 0.5 foot

altitude = 1 foot

thus, volume (V) = 1/3 πr²h

or, V = 1/3 π × (0.5)² × 1

or, V = 0.262 feet³

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Answer: y = 1/4x + -5/4

Step-by-step explanation:

Step 1: add -4x to both sides

step 2:  Divide both sides by -16.

Answer:

Slope is [tex]\frac{1}{4}[/tex]

Step-by-step explanation:

Rearranging the equation:

[tex]4x - 20 = 16y[/tex]

The coefficient of y needs to be '+1' for the above equation to be considered as the slope-intercept form:

Dividing both sides of the equation by '16':

[tex]\frac{16}{16}y = \frac{1}{16}(4x -20)[/tex]

[tex]y = \frac{4}{16}x - \frac{20}{16}[/tex]

Divide the numerator and denominator present in the coefficient of x  by the Highest Common Factor '4'. Also divide the numerator and denominator present in the coefficient of 'non-x' term by the Highest Common Factor '4':

[tex]y = \frac{1}{4}x - \frac{5}{4}[/tex]

The simplified form of the slope-intercept of the equation has been achieved. The coefficient of x is the slope of the graph:

m = Slope of graph =

sum of roots is 0 , -5 , 4

product of roots is -9, 25/4, 8

Discriminant = 36, 0 , - 16

An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax² + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term.

When a quadratic equation is solved, the two values of x that result are known as the roots of the equation. The zeros in the equation are another name for these quadratic equation roots. For instance, x = -1 and x = 4 are the roots of the equation x² - 3x - 4 = 0 since they each fulfill it.

The easiest method for locating a quadratic equation's roots is the quadratic formula. There are some quadratic equations that are difficult to factor, and in these cases we can quickly discover the roots by using the quadratic formula. The quadratic equation's roots also aid in determining the quadratic equation's sum of roots and product of roots. The quadratic formula's two roots are shown as a single statement.

The equation are :

x² - 9 = 0 ⇒ (x-3)(x+3) = 0 ,

sum of roots is 0

product of roots is -9

Discriminant = 0²  - 4*3(-3) = 36

The second equation :

4x² + 25 = - 20x ⇒ 4x² + 20x + 25 = 0 ⇒ 4x² + 10x + 10x + 25 ⇒2x(2x + 5) +5(2x+5) = 0  ⇒ (2x +5)²

sum of roots is -5

product of roots is 25/4

Discriminant = 0

x²/2 - 2x + 4 = 0 ⇒ x² - 4x + 8 = 0

Sum of roots 4

Product of roots 8

Discriminant = (-4)² - 4*1*8 = 16 - 32 = -16

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The bird is flying at a height of 404.9 feet above the ground

An equation is an expression that contains numbers and variables linked together by mathematical operations of addition, subtraction, multiplication, division and exponents. An equation can either  be linear, quadratic, cubic, depending of the degree of the variable.

Trigonometric ratio is used to show the relationship between the sides and angles of a right triangle.

Let h represent the height of the bird.

Using trigonometric ratio:

tan(39) = h / 500

h = 500 * tan(39)

h = 404.9 feet

The height of the bird is 404.9 feet

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Greetings!

Let's find the domain of the function.

We were given: f(x) = 9 - 5/(x^4)

The domain of a function is a group of values that are valid within the function. The domain could range from (-∞, ∞)

However, since we have a fraction, there will be a limit to the range of values.

In this case, we will have to find what value the variable "x" can't be.

Since we are analyzing the "5/(x^4)" part of the function, we know that fractions can't have a denominator of 0. This means that the denominator can't equal zero.

x^4 ≠ 0

Solve for "x" by taking the root of both sides of the equations.

x ≠ 0 Therefore x can't equal 0.

This means that the value of "0" will not be in our domain, making our domain from:

(-∞, 0)∪(0, ∞)

The above means that we are including all the values (due to the use of infinity), except for the value 0.

Answer:

(-∞, 0)∪(0, ∞)

The two-columns method can be used to prove that the triangles ΔABC and ΔADC are congruent (ΔABC ≅ ΔADC), by completing the included table as follows;

Step [tex]{}[/tex]                          Statement                  [tex]{}[/tex]              Reason

1  [tex]{}[/tex]                              [tex]\overline{AD}[/tex] ≅ [tex]\overline{AB}[/tex]   [tex]{}[/tex]                            

 [tex]{}[/tex]                                [tex]\overline{AB}[/tex] ⊥ [tex]\overline{BC}[/tex]          [tex]{}[/tex]                        Given

[tex]{}[/tex]                                  [tex]\overline{AD}[/tex] ⊥ [tex]\overline{DC}[/tex]                                

2. [tex]{}[/tex]                ΔABC and ΔADC are right Δs              Definition

3. [tex]{}[/tex]       [tex]\overline{AC}[/tex] is the hypotenuse side of the right Δs   Definition

4.     [tex]{}[/tex]                         [tex]\overline{AC}[/tex] ≅ [tex]\overline{AC}[/tex]                                   Reflexive property

5. [tex]{}[/tex]                          ΔABC ≅ ΔADC                            HL congruency rule

Congruent triangles are triangles in which the lengths of the three sides of each triangle are the same measure with the lengths of the three sides of the triangle with which they are congruent.

The details of the reasons used to prove the congruency of the triangles ΔABC and ΔADC, are presented as follows;

Definition of right triangles

A right triangle is a triangle that has an interior angle of 90°. The angle formed between two perpendicular lines is 90°.

The sides [tex]\overline{AB}[/tex] and [tex]\overline{BC}[/tex] of the triangle ΔABC are perpendicular, thereforte, the angle ∠B is 90°

Similarly, the sides [tex]\overline{AD}[/tex] and [tex]\overline{DC}[/tex] are perpendicular, therefore, the measure of the angle D, m∠D is 90°

Therefore, the triangles ΔABC and ΔADC are right triangles

Hypotenuse side of a right triangle

The hypotenuse side of a right triangle is the side linking the perpendicular sides and facing the 90° angle

HL congruency rule

The Hypotenuse-Leg (HL) congruency rule states that if the hypotenuse side and a leg of a right triangle are congruent to a hypotenuse side and a leg of a right triangle, the two right triangles are congruent.

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The length and the width are 12 yards and 16 yards

The formula for calculating the perimeter of a rectangle is expressed as;

Perimeter =2( l + w)

Given that the parameter are;

Length = 4 + w

Substitute the values into the formula

52 = 2( 4 + w + w)

collect like terms

52 = 2(4 + 2w)

expand the bracket

52 = 8 + 4w

collect like terms

4w = 52 - 8

subtract the values

4w = 48

Make 'x' the subject of formula

w = 12 yards

L = 16 yards

Hence, the values are 12 yards and 16 yards

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For the given grid a. The area of the triangle is 21 square units. b. The area of the rectangle is 42 square units.

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.

Given that the points on the grid are 1 units apart we have:

The height of the triangle is 6 units and the base of the triangle is 7 units.

The area of the triangle is:

A = 1/2 (b)(h)

A = 1/2(7)(6)

A = 21 square units.

The area of the triangle is 21 square units.

The length of the rectangular grid is 7 units and the width is 6 units.

The area of the rectangle is:

A = (l)(b)

A = (7)(6)

A = 42 square units.

The area of the rectangle is 42 square units.

Hence, a. The area of the triangle is 21 square units. b. The area of the rectangle is 42 square units.

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The three subcategories in the previous scratch list are high-end linens, affordability, and food and drink-free snacks.

High-end linens include double-thick bath towels and 600-thread-count sheets. Affordability includes $100/night for four-star rooms. Food and drink-free snacks include coffeemakers and selected teas.

This scratch list provides guests with luxurious amenities and convenience. High-end linens enhance the comfort of a room, while affordability makes it accessible to many.

Food and drink-free snacks add to the convenience of a stay by providing guests with refreshments. All of these amenities contribute to a pleasant and comfortable stay.

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The amount company will make in 6 months is $24327.43.

The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest. Compound interest is commonly abbreviated C.I. in mathematics.

Given the interest rate = 12.5% = 0.125

principal  = $12,000

time period = 6 months

CI is given by,

CI = [tex]p(1 + \frac{r}{n})^{nt}[/tex]

n = 1

substitute the values,

A  = 12000(1 + (0.125/1))¹ˣ⁶

A = 12000(1.125)⁶

A = 24327.43

A = $24327.43

Hence the amount will be $24327.43.

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The area of the polygon is  207.06 unit².

How to find the Area of the Polygon with n-sides?

For finding the area of a polygon which is not regular or its formula is not defined, we split the figure into triangles, squares, trapezium, etc. The purpose is to visualize the given geometry as a combination of geometries for which we know how to calculate the area. We then calculate the area for each of the part and then add them up to obtain the area of the polygon.

Here, we have given a polygon in the combination of a square and a triangle.

So, to find an area of the polygon we need first to find the area of square and triangle.

Area of Polygon Formulas:

Name of the Polygon Area Formula

Triangle = 1/2 × base × height

Square = (side)²

Area of Square = (13)²

= 169 unit².

Area of Triangle = 1/2 × base × height

To find base:

(height)² + (base)² = (hypotenuse)²

(13)² = (11)² + (base)²

169 = 121 + (base)²

(base)²  = 169 - 121

base = √48 = 6.9

Area of Triangle = 1/2 × (6.92) × 11

= 38.06 unit².

Area of Polygon = Area of Square  + Area of Triangle

=  169 + 38.06

= 207.06 unit².

Therefore, the area of the polygon is  207.06 unit².

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The order of the numbers from least to greatest is -3, √3, 3, 3².

√3  is an irrational number

Arranging numbers in order from least to greatest is the arrangement of numbers in such a way that the smallest number comes first and the highest number comes last.

We have 3, 3², √3, -3

where 3² = 9 and √3 = 1.73

Thus, the order from least to greatest will be:

-3, √3, 3, 3²

Irrational numbers are mostly in the form of square roots, which when evaluated cannot be expressed in the form a/b e.g √7, π, 2/√5, 3√11, etc.

Thus, √3  is irrational

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The relative frequency of an event is a way to estimate the probability of that event happening.

For example, if you spin a spinner 50 times and the spinner lands on red 20 times, the relative frequency of getting red would be 20/50 = 0.4. This is a statistical estimate of the probability of the spinner landing on red, and it's based on the limited number of trials (spinning the spinner 50 times). As you increase the number of trials, the relative frequency will become a more accurate estimate of the true probability of the event. So if you spin the spinner 100 times instead of 50 times, and the spinner lands on red 40 times, the relative frequency would be 40/100 = 0.4. This shows that as the number of trials increases, the relative frequency becomes a better estimate of the true probability of the event. The reason for this is that the more trials you have, the more representative your sample is of the population, and the closer your estimate will be to the true probability of the event. In this case, the true probability of the spinner landing on red is unknown, but as you spin it more times, the relative frequency will approach the true probability, providing a more accurate estimate.

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Answer:

0.615, 0.609, 0.516, 0.506

Answer:

4e-4=e-4

subtract 4 from both sides

4e=e

subtract e from both sides

4e-e=0

4e-e equals 3e

3e= 0

divide by 3 both sides

3/3=0/3

e=0

Answer:

e = 0

Step-by-step explanation:

4e - 4 = e - 4 ( subtract e from both sides )

3e - 4 = - 4 ( add 4 to both sides )

3e = 0 ( divide both sides by 3 )

e = 0 ÷ 3 = 0

Answer:

in this case, the polynomial function has a degree of 5 which means that is has at most 5 complex zeros. however, it doesn't necessarily mean it will have 5 zeros, it could have fewer or none at all.

The required time taken by the Devin to complete 5.8 laps is 24.36 minutes.

To find out how long it takes Devin to complete 5.8 laps, we need to multiply the time it takes to complete one lap by the number of laps. If it takes Devin 4.2 minutes to walk a lap, then to complete 5.8 laps, it would take.

time = 4.2 minutes/lap * 5.8 laps

= 24.36 minutes.

So, it would take Devin 24.36 minutes to complete 5.8 laps.

This information can be useful for determining Devin's walking pace, comparing his time to that of others, or calculating the total distance he has covered. It's important to note that this calculation assumes that Devin's pace remains constant throughout the entire walk, which may not always be the case.

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