A rectangular room is tiled with tiles 1
2
inches long and 1
2
inches wide. If the room is L tiles long and W tiles
wide, find the room’s area, in square inches.

The correct option is D.

Given that ABCD is a parallelogram, we need to prove AD ≅ CB.

So,

Properties of a parallelogram =

Opposite sides are parallel.

Opposite sides are congruent.

Opposite angles are congruent.

             Statement                                                    Reason

1. ABCD is a parallelogram                                           Given

2. AB║CD and AD║BC                                      Definition of parallelogram

3. ∠BAC ≅ ∠DCA and ∠BCA ≅ ∠DAC    Alternate interior angles theorem

4. AC ≅ AC                                                              Reflexive property

5. ΔBAC ≅ ΔDCA                                                       A.S.A Postulate

6. AD ≅ CB                                                                        CPCTC

Hence the correct option is D.

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The value of the angle csc θ of the given problem is: 5/4

There are different trigonometric ratios such as:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

Now, we are given that:

cot θ = 4/3

We know that cot θ = 1/tan θ

Thus:

tan θ = 3/4

Since θ is an acute angle, it means it is less than 90 degrees. Thus, using Pythagoras theorem, we can find the third side which is the hypotenuse to be 5

Now, csc θ = 1/sin θ

Thus: sin θ = 4/5

csc θ = 5/4

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To solve the expression [tex]\large\sf\:\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\\[/tex] given the conditions [tex]\large\sf\:3^a = 9^b = 27^c\\[/tex], we can use logarithmic properties and the fact that [tex]\large\sf\:3^2 = 9\\[/tex] and [tex]\large\sf\:3^3 = 27\\[/tex].

Let's start by finding the values of a, b, and c using logarithmic properties:

Taking the logarithm of both sides of [tex]\large\sf\:3^a = 9^b\\[/tex], we get:

[tex]\large\sf\:\log_3(3^a) = \log_3(9^b)\\[/tex]

Applying the power rule of logarithms, we can bring down the exponents:

[tex]\large\sf\:a\log_3(3) = b\log_3(9)\\[/tex]

Since [tex]\large\sf\:\log_3(3) = 1[/tex] and [tex]\large\log_3(9) = 2\\[/tex], we simplify to:

[tex]\large\sf\:a = 2b\\[/tex] ---- (1)

Similarly, taking the logarithm of both sides of [tex]\sf\:9^b = 27^c\\[/tex], we get:

[tex]\large\sf\:b\log_3(9) = c\log_3(27)\\[/tex]

Using the values of [tex]\sf\:\log_3(9)\\[/tex] and [tex]\sf\:\log_3(27)\\[/tex] as before, we have:

[tex]\large\sf\:b(2) = c(3)\\[/tex]

Simplifying, we get:

[tex]\large\sf\:2b = 3c\\[/tex] ---- (2)

Now, let's substitute the value of b from equation (1) into equation (2):

[tex]\large\sf\:2(2b) = 3c\\[/tex]

[tex]\large\sf\:4b = 3c\\[/tex]

Rearranging, we find:

[tex]\large\sf\:c = \frac{4b}{3}\\[/tex] ---- (3)

We now have expressions for a, b, and c in terms of b. Let's substitute these into the expression [tex]\large\sf\:\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\\[/tex]:

[tex]\large\sf\:\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = \frac{2b}{b} + \frac{b}{\frac{4b}{3}} + \frac{\frac{4b}{3}}{2b}\\[/tex]

Simplifying further, we get:

[tex]\large\sf\:\frac{2}{1} + \frac{3}{4} + \frac{2}{3}\\[/tex]

Finding the common denominator and combining the fractions, we have:

[tex]\large\sf\:\frac{24}{12} + \frac{9}{12} + \frac{8}{12}\\[/tex]

Adding the fractions together, we obtain:

[tex]\large\sf\:\frac{24 + 9 + 8}{12} = \frac{41}{12}\\[/tex]

Therefore, [tex]\large\sf\:\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = \frac{41}{12}\\[/tex].

The values of the side m and n for the right triangle are 36 and 12√3 respectively using the trigonometric ratio of sine

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

recall that sin60° = √3/2 and sin30° = 1/2

sin 60° = m/(24√3) {opposite/hypotenuse}

√3/2 = m/(24√3)

m = (24√3 × √3)/2 {cross multiplication}

m = 12 × 3

m = 36

sin 30° = n/(24√3)

1/2 = n/(24√3)

n = 24√3/2 {cross multiplication}

n = 12√3

Therefore, the values of the side m and n for the right triangle are 36 and 12√3 respectively using the trigonometric ratio of sine

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Answer:10

Step-by-step explanation: You plug in 3 for x so you get the equation 2(3)+4

Which becomes 6+4

Which equals 10

Answer:

$3827.00

Step-by-step explanation:

To calculate the total compound amount at the end of six years with an interest rate of 1.5% compounded annually, we can use the formula for compound interest:

A = P * (1 + r/n)^(n*t)

Where:

A = Total compound amount

P = Principal amount (initial investment)

r = Annual interest rate (as a decimal)

n = Number of times interest is compounded per year

t = Number of years

In this case:

P = $3500

r = 1.5% = 0.015 (as a decimal)

n = 1 (compounded annually)

t = 6 years

Substituting these values into the formula, we have:

A = 3500 * (1 + 0.015/1)^(1*6)

Calculating this expression, we get:

A ≈ 3500 * (1.015)^6 ≈ 3500 * 1.093717 ≈ 3827.00

Rounding the total compound amount to 2 decimal places, the final value is approximately $3827.00

The true statements about the graph of h(x) = ∛(x-4) are:

The domain of h(x) is the set of all real numbers.

For all points (x, h(x)), h(x) exists if and only if x - 4≥0

The graph of h(x) intercepts the x-axis at (4, 0).

The domain of h(x) is the set of all real numbers since there are no restrictions on the input variable x.

For all points (x, h(x)), h(x) exists if and only if x - 4 ≥ 0. This is because the cube root function is only defined for non-negative values, and the expression inside the cube root, x - 4, must be non-negative for h(x) to exist.

The graph of h(x) intercepts the x-axis at (4, 0) since plugging in x = 4 into the equation gives h(4) = ∛(4-4) = ∛0 = 0. This means that the point (4, 0) lies on the graph.

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There are 1365 different ways for Andre to select the blue fabrics for his quilt.

To determine the number of different ways Andre can select the blue fabrics for his quilt, we can use the concept of combinations.

Andre wants to select 4 different fabrics from a pool of 15 options for the blue portion of the quilt. This can be calculated using the formula for combinations:

C(n, r) = n! / (r! × (n-r)!)

where n is the total number of options and r is the number of selections.

In this case, we have:

n = 15 (number of options for blue fabrics)

r = 4 (number of fabrics to be selected)

Plugging in these values into the combination formula, we get:

C(15, 4) = 15! / (4! × (15-4)!)

Simplifying the equation, we have:

C(15, 4) = 15! / (4! × 11!)

Now, let's calculate the factorial terms:

15! = 15 × 14 × 13 × 12 × 11!

4! = 4 × 3 × 2 × 1

11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Substituting these factorial terms into the equation:

C(15, 4) = (15 × 14 × 13× 12 × 11!) / (4 × 3 × 2 × 1 × 11!)

Simplifying further:

C(15, 4) = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1)

C(15, 4) = 1365

Therefore, there are 1365 different ways for Andre to select the blue fabrics for his quilt.

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The The equation that would represent Maria's possible sales, S(c, b), is:

S(c, b) = 1.50c + 1.20b

The term 1.50c represents the total revenue from selling bags of chips.

The term 1.20b represents the total revenue from selling candy bars.

So, the equation can be written as

S(c, b) = 1.50c + 1.20b

This equation represents the total sales amount (S) based on the quantities of bags of chips (c) and candy bars (b) sold.

The equation calculates the sales by multiplying the number of bags of chips (c) by their price of $1.50 each and the number of candy bars (b) by their price of $1.20 each.

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The distance between the side AB are 314.2 ft.

In a triangle ABC AC = 200 ft, m ∠A = 72°, and m ∠B = 37°. we need to find the measure of the side AB,

So,

Using sine law.

Sin A / CB = Sin B / AC = Sin C / AB

Using the angle sum property of a triangle we find the value of angle C,

∠C = 180° - (∠A + ∠B)

∠C = 180° - (72° + 37°)

∠C = 180° - 109°

∠C = 71°

So,

Sin 37° / 200 = Sin 71° / AB

AB = 200 / Sin 37° × Sin 71°

AB = 314.2 ft

Hence the distance between the side AB are 314.2 ft.

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Answer: To find the sales tax rate as a percentage, we can use the following formula:

Sales Tax Rate = (Sales Tax / Cost Before Tax) * 100%

In this case, the sales tax is given as $20, and the cost before tax is $500. Plugging these values into the formula, we have:

Sales Tax Rate = ($20 / $500) * 100%

Simplifying the expression:

Sales Tax Rate = (0.04) * 100%

Sales Tax Rate = 4%

Therefore, the sales tax rate for the item is 4%.

Step-by-step explanation:

Answer:

Step-by-step explanation:

For the package of 10 folders: Unit price = $1.20 / 10 = $0.12 For the package of 8 folders: Unit price = $1.12 / 8 = $0.14

Answer:

See below

Step-by-step explanation:

Where the graph crosses the x-axis will solve the equation

The value of x in the given right triangle HJK include the following: C. 10.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the side lengths are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side):

Hypotenuse, x = 5 × 2

Hypotenuse, x = 10.

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The correct statements regarding the domain and the range of the functions are given as follows:

For the square root function, we have that:

Hence if we multiply the function by a negative number, the domain of the function remains constant, but the range is changed.

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Answer:

The answer would lie within 31 degrees of MP and also as in PM.

Answer:

central m arc MP=118°

Step-by-step explanation:

here

central m arc MN=2* inscribed m arc MN=2*31=62°

again

central m arc MN+ central m arc MP=180° being linear pair

substituting value

62°+central m arc MP=180°

central m arc MP=180°-62°

central m arc MP=118°

The numerical value of x in the dimensions of the rectangle is 9.

A rectangle is a 2-dimensional shape with parallel opposite sides equal to each other and four angles are right angles.

Area of a rectangle is expressed as;

A = length × width

From the diagram:

Wdith of the rectangle = x - 2

Length of the rectangle = x + 5

Area = 98 ft²

Since area of a rectangle =  length × width:

98 = ( x - 2 )( x + 5 )

Simplify

( x - 2 )( x + 5 ) = 98

Apply distributive property:

x² + 3x - 10 = 98

x² + 3x - 10 - 98 = 0

x² + 3x - 108 = 0

Factor using AC method:

( x - 9 )( x + 12 ) = 0

Hence:

x - 9 = 0 or x + 12 = 0

x = 9 or x = -12

Since, we are dealing with dimensions, we take the positive value.

Hence, the value of x is 9.

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The graph most often used to show change in data across time is the line graph.

The graph most often used to show change in data across time is the line graph. A line graph is an effective visualization tool that displays data points as a series of connected data markers, forming a line.

It is commonly used to illustrate trends, patterns, or fluctuations in data over a continuous time interval.

The x-axis represents time, while the y-axis represents the variable being measured. By plotting data points and connecting them with lines, line graphs provide a clear visual representation of how the data changes over time, allowing for easy identification of trends, seasonality, growth, or decline in the data series.

Line graphs are widely utilized in various fields, including economics, finance, science, and social sciences, to present temporal data in a comprehensive and understandable manner.

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It will take approximately 2.32 hours to fill 11 tanks.

First, we need to convert the capacity of the tanks from gallons to liters, as the rate at which we empty the bottle is given in liters. Since 1 gallon is equal to 3.8 liters, the capacity of each tank is 50 gallons * 3.8 liters/gallon = 190 liters.

Now, let's calculate the time it takes to fill one tank:

The bottle is emptied at a rate of 2 liters every 8 seconds. To find the time it takes to fill one tank, we divide the capacity of the tank (190 liters) by the rate at which we empty the bottle (2 liters/8 seconds):

Time for one tank = 190 liters / (2 liters/8 seconds) = 190 liters * (8 seconds/2 liters) = 760 seconds.

Next, we need to find the total time required to fill 11 tanks. Since each tank takes 760 seconds to fill, we multiply this by 11 to account for the 11 tanks:

Total time to fill 11 tanks = 760 seconds/tank * 11 tanks = 8,360 seconds.

Finally, we convert the total time from seconds to hours. There are 60 seconds in a minute and 60 minutes in an hour, so:

Total time to fill 11 tanks = 8,360 seconds * (1 minute/60 seconds) * (1 hour/60 minutes) = 2.32 hours (rounded to two decimal places).

Therefore, it will take approximately 2.32 hours to fill 11 tanks.

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When a student is selected at random, the solution for the various possible outcome would be given below as follows:

p(pass or did not study)=0.409

p(pass or did study)=0.312

p(fail and pass)= 0.201

p(pass or fail)= 1

The formula that is used to calculate probability = possible outcome/sample space.

The total number of scores of the statistics students = 93

For p(pass or did not study):

possible outcome = 38

sample size = 93

probability = 38/93 = 0.409

For p(pass or did study):

possible outcome = 29

sample size = 93

probability = 29/93 = 0.312

For p(fail and pass):

p(fail) = 26/93

p(pass) = 67/93

probability = 26/93× 67/93

= 1742/8649

= 0.201

For p(pass or fail);

P(pass) = 67/93

P(fail) = 26/93

Probability of pass or fail = 67/93+26/93

= 93/93 = 1

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The expression that is undefined in the set of real numbers would be = 0/-4. That is option B.

Real numbers are those numbers that can either be positive or negative and that are rational or irrational numbers which are different from complex numbers.

An expression can be said to be an undefined set of real numbers when it's numerator = 0 as seen in 0/-4.

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Answer:

6 m, 24 m

Step-by-step explanation:

area of kite = (diagonal 1 × diagonal 2) / 2

Let the length of the shorter diagonal = x.

The length of the longer diagonal is 4x.

The area is the product of the diagonals divided by 2.

area = 4x × x / 2 = 4x²/2 = 2x²

The area is 72 m².

Therefore, 2x² must equal 72 m².

2x² = 72 m²

Divide both sides by 2.

x² = 36 m²

Take the square root of both sides.

x = 6 m

The short diagonal has length 6 m.

The long diagonal is 4x long.

4x = 4(6 m) = 24 m

Answer: 6 m, 24 m

Answer:

6 metres and 24 metres

Step-by-step explanation:

The kite will be composed of two smaller triangles and two larger triangles.

Let's call the shorter diagonal (across the kite) W.

So the longer diagonal (top to bottom of kite) will be 4W.

area = (L X W)/2

= (4W X W)/2

= 4W²/2

= 2W².

2W² = 72

W² = 36

W = 6 metres.

so the shorter diagonal is 6 metres.

the longer diagonal is 4 X 6 = 24 metres

The missing angle is <B= 40 degree and missing side length is AB = 12.25 and AC = 19.068.

To find the missing angle and side measures of ΔABC, we can use the properties of a triangle.

Given:

∠A = 50°

∠C = 90°

CB = 16

We can start by finding the measure of ∠B:

∠A + ∠B + ∠C = 180° (Sum of angles in a triangle)

50° + ∠B + 90° = 180°

∠B + 140° = 180°

∠B = 180° - 140°

∠B = 40°

Now, using Sine law

CB/ sin A = AB / sin C

16 / sin 50 = AB / sin 90

16 / 0.766044 = AB

AB = 12.25

Again 12.25 = AC/ sin B

12.25 = AC / sin 40

AC = 19.068

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The correct system of inequalities is 1500x + 1250y ≥ 175,000 and x + y ≥ 200, option A is correct.

We have two carriers: land carriers and air carriers.

The land carriers can transport 1500 pieces of mail per load, and the air carriers can transport 1250 pieces of mail per load.

The loading equipment at the distribution center can handle up to 200 loads per day.

To represent this situation using a system of inequalities, we can define the variables x and y as the number of loads by land carriers and air carriers, respectively.

The first inequality, 1500x + 1250y ≥ 175,000, represents the total number of pieces of mail transported per day.

The second inequality, x + y ≥ 200, represents the total number of loads.

Hence,  the correct system of inequalities is 1500x + 1250y ≥ 175,000 and x + y ≥ 200.

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The linear function equation, y = m·x + c, indicates;

6. The growth in the y-values over each unit x-value interval is a difference of m units.

Part I. Please find attached the graph of y = x + 3, created with MS Excel, which shows that the differences between the y-values is 2 units over each x-value interval of 2 units.

a) The difference between the x-values is; k

The difference between the y-values is; m·k

The difference in the y-value is; m·k

Please find attached the completed tables in the following section

The equation of a linear function is; y = m·x + c

Where;

m = The slope of the graph = (y₂ - y₁)/(x₂ - x₁)

(x₁, y₁), and (x₂, y₂), are ordered pair of data on the graph of the linear function

c = The y-intercept of the linear function

(y₂ - y₁)/(x₂ - x₁) = m

(y₂ - y₁) = m × (x₂ - x₁)

Δy = m × Δx

When Δx = 1, we get;

Δy = m

Therefore, the y-value of a linear function increases by m, the slope value, when the x-value increases by 1, which indicates that the linear function grows by equal differences over the same interval of the input value of the function

Part I; Please find a graph of the linear function, y = 1·x + 3, which shows that each equivalent interval of 2 units in the x-axis, produces a difference in the y-value or y-axis of 2 units.

a) Considering the interval p and p + k, let y₁ represent the y-value at p and let y₂ represent the y-value at p + k, we get;

Slope, m = (y₂ - y₁)/((p + k) - p) = (y₂ - y₁)/k  

Therefore;

(y₂ - y₁)/k = m

(y₂ - y₁) = m × k

Therefore, the increase in the y-value, for an increase in the x-value of k is a constant,  m·k =  (y₂ - y₁)

Therefore the difference in the y-value for the specified x-values is a constant m·k

The possible equation in the question obtained from a similar question on the website is; y = -(1/2)·x + 10

The completed tables are;

[tex]\begin{tabular}{ | c | c | c | c | }\cline{1-4}& Interval; p = 2& p+ k = 6&\\ \cline{1-4}x-value & 2 & 6 & 4 \\\cline{1-4}y-value & y = (-1/2)\cdot (2)+10 =9 & y = (-1/2)\cdot (6)+10=\underline{7} & -2 \\\cline{1-4}\end{tabular}[/tex]

[tex]\begin{tabular}{ | c | c | c | c | }\cline{1-4}& Interval; p = 10& p+ k = 14&\\ \cline{1-4}x-value & 10 & 14 & 4 \\\cline{1-4}y-value & y = (-1/2)\cdot (10)+10 =\underline{ 5 }& y = (-1/2)\cdot (14)+10=3 & -2 \\\cline{1-4}\end{tabular}[/tex]

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Answer:

25%

Step-by-step explanation:

9+3=12 dance people total

divide 3 by 12 to determine debate members over dance members

=.25 or 25%

The probability that a customer ordered vanilla ice cream given they ordered a sundae is 0.75, or 75%.

To find the probability that a customer ordered vanilla ice cream given they ordered a sundae, we can use the concept of conditional probability.

The conditional probability of an event A given event B is denoted as P(A|B) and is calculated as the probability of both events A and B occurring divided by the probability of event B.

In this case, we want to find P(vanilla|sundae), which represents the probability of a customer ordering vanilla ice cream given that they ordered a sundae.

Using the given probabilities:

P(vanilla) = 0.3

P(sundae) = 0.2

P(vanilla and sundae) = 0.15

The probability of a customer ordering a sundae is the denominator for the conditional probability. Therefore, P(sundae) will be the denominator.

P(vanilla|sundae) = P(vanilla and sundae) / P(sundae)

Substituting the given values:

P(vanilla|sundae) = 0.15 / 0.2

Now we can simplify:

P(vanilla|sundae) = 0.15 / 0.2

= 0.75

Therefore, the probability that a customer ordered vanilla ice cream given they ordered a sundae is 0.75, or 75%.

In conclusion, based on the given probabilities, there is a 75% chance that a customer who ordered a sundae also ordered vanilla ice cream.

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The area of the rhombus is 100 mm². The Option D.

To get area of a rhombus, we will use the formula: Area = (d₁ * d₂) / 2 where d₁ and d₂ are the lengths of the diagonals.

Given that the horizontal diagonal length is 25 millimeters (d₁ = 25 mm) and the vertical diagonal length is 8 millimeters (d₂ = 8 mm.

We will substitute values into the formula:

Area = (25 mm * 8 mm) / 2

Area = 200 mm² / 2

Area = 100 mm².

Full question:

A rhombus with horizontal diagonal length 25 millimeters and vertical diagonal length 8 millimeters. what is the area of the rhombus? 25 mm2 33 mm2 50 mm2 100 mm2

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