Which graph shows a proportional relationship?

Answer choices attached.

Answer:

The ladder reaches 142 inches and the base of ladder is 25 inches far from the base of the wall.

Step-by-step explanation:

Let us draw a diagram for the given situation.

In the attached figure,

AB = the height of wall where the ladder reaches

AC = ladder

BC = distance of base of ladder from the base of the wall

In triangle ABC,

[tex]\sin C=\frac{AB}{AC}\\\\\sin80=\frac{AB}{12}\\\\AB=12\sin80\\\\AB=11.82[/tex]

And

[tex]\cos C=\frac{BC}{AC}\\\\\cos80=\frac{BC}{12}\\\\BC=12\cos80\\\\BC=2.08[/tex]

Now, 1 feet = 12 inches

Hence, 11.82 ft = 141.84 inches

and 2.08 feet = 24.96 inches

Therefore, in the nearest inches, we have

The ladder reaches 142 inches and the base of ladder is 25 inches far from the base of the wall.

Answer:

'The product of two rational numbers is rational'                        

Step-by-step explanation:

Statement 'The product of two rational numbers'.

We know that, 'The product of two rational numbers is rational'.

We can prove it,

Let [tex]\frac{a}{b}[/tex] and [tex]\frac{p}{q}[/tex] are two rational number.

Where, a,b,p,q are integers and [tex]b,q\neq 0[/tex]

Now, Product the two rational number

[tex]\frac{a}{b}\times \frac{p}{q}=\frac{ap}{bq}[/tex]

Since, ap and bq are integers and [tex]bq\neq 0[/tex]

Thus, [tex]\frac{ap}{bq}[/tex] is an fraction with integers in the numerator and denominator making it a rational number.

Therefore, 'The product of two rational numbers is rational' is true.

Answer: The median is the only appropriate measure of center.

Step-by-step explanation:

The statement is ''Both the median and mode are appropriate measures of center''. Actually, the three, that is, the mean, media and mode can be used to measure the central of tendency. But the MEAN has a problem, it can be Skewed by large values in the distribution/data. Assuming, the days of the week when patients visiting a dentist's office are;

Monday: 8, Tuesday: 41, Wednesday: 35, Thursday: 39, Friday: 36, and Saturday: 42.

And the mean of the distribution = 33.5. Most patients visit the dentist Tuesday to Saturday, where Saturday is the highest. The data is skewed. So, the MEDIAN will do the job perfectly.

For the mode; since we do not have; (1). two highest mode with the same value in the data provided and, (2). the highest value and the second to the highest values are not too far from each other. We can use the MODE here for measuring the central tendency.

===> CONCLUSION: (1). Assuming the data fails the two condition for mode, the most correct statement will be ''The median is the only appropriate measure of center".

===> (2). Also, for this data, the MEDIAN will be the only appropriate measure of center.

R measures 42.8 degrees, which is around average.

The properties of the triangle are:

1. The sum of all the angles of a triangle (of all types) is equal to 180°.

2. The sum of the length of the two sides of a triangle is greater than the length of the third side.

3. In the same way, the difference between the two sides of a triangle is less than the length of the third side.

The sine rule states that for a triangle with sides a, b, and c opposite to angles A, B, and C respectively,

a/sinA = b/sinB = c/sinC

Using this rule, we can find the measure of angle R as follows:

a = ST = 30

b = RT = 35

c = RS

Let angle R be opposite to side b.

sinR/sinS = b/c

sinR/sin69 = 35/c

c = 35/sin69 x sinR

Now we can use the sine rule to find c:

a/sinA = b/sinB = c/sinC

30/sin69 = 35/sinR = c/sin42

c = 35/sinR x sin42

Equating the two expressions for c, we get:

35/sin69 x sinR = 35/sinR x sin42

Simplifying this equation, we get:

sinR/sin69 = sin42

sinR = sin69 x sin42

sinR = 0.672

Taking the inverse sine of both sides, we get:

R = sin^{-1}(0.672)

R ≈ 42.8 degrees

Therefore, the measure of angle R is approximately 42.8 degrees.

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Answer:

The volume is 180 cube cm.

Step-by-step explanation:

By the given diagram,

The composite figure is made by the two cuboids,

First having dimensions, 2 cm × 5 cm × 6 cm,

And second having dimensions, 8 cm × 5 cm × 3 cm,

So, the volume of the given composite figure = the volume of first cuboid + the volume of second cuboid

= 2 × 5 × 6 + 8 × 5 × 3,

= 60 + 120

= 180 cube cm

The volume of the composite space figure to the nearest whole number is: 180 cube cm.

Here, we have,

from the given information, we get,

By the given diagram,

The composite figure is made by the two cuboids,

First having dimensions, 2 cm × 5 cm × 6 cm,

And second having dimensions, 8 cm × 5 cm × 3 cm,

So, the volume of the given composite figure

= the volume of first cuboid + the volume of second cuboid

= 2 × 5 × 6 + 8 × 5 × 3,

= 60 + 120

= 180 cube cm

Hence, the volume of the composite space figure to the nearest whole number is: 180 cube cm.

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Answer:

The density of the wood is, 0.64 [tex]g/cm^3[/tex]

Step-by-step explanation:

Using the density formula:

[tex]\text{density} = \frac{\text{mass}}{\text{Volume}}[/tex]         .....[1]

As per the statement:

If the volume of a cube of wood is 15.625 cm3 and its mass is 10.00 grams.

⇒Volume = 15.625 cubic cm and mass = 10 gram

Substitute the given values in [1] we have;

[tex]\text{density} = \frac{10}{15.625}[/tex]

Simplify:

density = 0.64 [tex]g/cm^3[/tex]

Therefore, the density of the wood is, 0.64 [tex]g/cm^3[/tex]

Answer:

a

Step-by-step explanation:

The next number in pattern 1/324, 1/108, 1/36, 1/12 would be 1/4

Firstly , let us learn about types of sequence in mathematics.

Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.

[tex]\boxed{T_n = a + (n-1)d}[/tex]

[tex]\boxed{S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

d = common difference between adjacent numbers

Geometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ration.

[tex]\boxed{T_n = a ~ r^{n-1}}[/tex]

[tex]\boxed{S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

r = common ratio between adjacent numbers

Let us now tackle the problem!

Given:

This problem is about Geometric Sequence. Let's see why.

[tex]\frac{1}{324} , \frac{1}{108} , \frac{1}{36} , \frac{1}{12} , . . .[/tex]

T₁ = [tex]\frac{1}{324}[/tex]

T₂ = [tex]\frac{1}{108}[/tex]

T₃ = [tex]\frac{1}{36}[/tex]

T₄ = [tex]\frac{1}{12}[/tex]

[tex]\large {\boxed {\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = r = 3} }[/tex]

From the above results it can be concluded that this is Geometric Sequence.

The next number will be the fifth term and can be calculated as following.

[tex]T_n = a ~ r^{n-1}[/tex]

[tex]T_5 = \frac{1}{324} \times 3^{5 - 1}[/tex]

[tex]T_5 = \frac{1}{324} \times 3^4[/tex]

[tex]T_5 = \frac{1}{324} \times 81[/tex]

[tex]\large {\boxed {T_5 = \frac{1}{4} } }[/tex]

Grade: Middle School

Subject: Mathematics

Chapter: Arithmetic and Geometric Series

Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term

It will take at least 12 years for Sarah's $600 deposit at a 2.5% annual interest rate to grow to more than $800, assuming the interest is compounded annually.

To determine how long it will take for Sarah's $600 deposit at a 2.5% annual interest rate to grow to over $800, we can use the formula for compound interest: [tex]A = P(1 + r/n)^{(nt)[/tex], where:

Since the problem doesn't specify the compounding frequency, let's assume it's compounded annually for simplicity:
[tex]A = 600(1 + 0.025)^t[/tex]. We need A to be greater than $800.

To find the minimum number of years (t), we solve for t:

[tex]800 < 600(1 + 0.025)^t[/tex]

Solving for t gives us: t > ln(800/600) / ln(1.025).

Using a calculator, we find that t > 11.89.

Therefore, it will take at least 12 years for the amount in Sarah's account to exceed $800.

a) 116 ways to select seven people.

b) ways to select 7 women.

c) The probability is 0.004662

Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant. You can choose the components of combos in any order. Permutations and combinations can be mixed up.

a) Number of ways can seven people be selected from this group of thirteen

C( 13, 7)

= 13! /6! 7!

= ​1716

b) Number of ways seven women selected from  Eight women

C(8 ,7)

= 8!/ 7! 1!

= 8

c) The probability that the selected group will consist of all women

= 8 / 1716

= 0.004662

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Answer:

x³y³ is monomial.

Step-by-step explanation:

Given :Polynomial  x³y³ .

To find : Identify monomial binomial trinomial.

Solution : We have given that

Polynomial  x³y³ .

Monomial : A polynomial with just one term.

Examples : 1) 3xy

2) 5y²x³  Both are monmial .

x³y³ is also a monomial because it has only one term.

Therefore,  x³y³ is monomial.

In geometry, a chord, radius, diameter, tangent, secant, and the point of tangency are all parts related to a circle. A radius can also be an axis, a minor arc is the smaller arc between two points on a circle, and a major arc is the larger one. These terms help in understanding the properties and relations of lines to a circle.

In the context of circles in geometry, various terms define specific parts or aspects of a circle. Here's what each term represents:

A minor arc is the shortest arc connecting two endpoints on a circle, and a major arc is the longer arc connecting the same two endpoints. The length of an arc is proportional to the size of its associated angle at the center of the circle.

When drawing or labeling the parts of a circle, it's important to note these definitions. For instance, if given a diagram with a point on the circle labeled 'A' and the center labeled 'O', the segment OA would be a radius. If there's a straight line that touches the circle at point A and doesn't cross through it, that would be your tangent, and point A is the point of tangency. Any straight line through A that crosses through another point on the circle, say, point B, would be a secant, and the line OAB (if extended through O) would be a diameter. The arc's size determines if it is a minor or major arc, depending on whether it is more or less than half the circle's circumference.

The equation of line passes through the points (2, -1) and (5, -10) will be;

⇒ y = - 3x + 5

What is Equation of line?

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

Two points on the line are (2, -1) and (5, -10)

Now,

Since, The equation of line passes through the points (2, -1) and (5, -10)

So, We need to find the slope of the line.

Hence, Slope of the line is,

m = (y₂ - y₁) / (x₂ - x₁)

m = (- 10 - (-1)) / (5 - 2)

m = - 9 / 3

m = - 3

Thus, The equation of line with slope - 3 is,

⇒ y - (-1)= - 3 (x - 2)

⇒ y + 1 = - 3x + 6

⇒ y = - 3x + 6 - 1

⇒ y = - 3x + 5

Therefore, The equation of line passes through the points (2, -1) and

(5, -10) will be;

⇒ y = - 3x + 5

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Answer:

∠C = 107°

Step-by-step explanation:

Quadrilateral ABCD is inscribed in a circle, and a quadrilateral inscribed in a circle is known as cyclic quadrilateral.

So, ABCD is cyclic quadrilateral.

∠A = 2x + 3

∠B = 2x - 4

∠D = 3x + 9

Also. the sum of opposite angles of the cyclic quadrilateral is supplementary.

⇒ ∠B + ∠D = 180

⇒ 2x - 4 + 3x + 9 = 180

⇒ 5x = 175

⇒ x = 35

Now, ∠A = 73

Also, ∠A + ∠C = 180

⇒ ∠C = 180 - 73

⇒ ∠C = 107°

Answer:

Let's consider this right triangle described in the picture.

∠ABD = ∠CBD, since BD is a bisector.

Angles ∠ADB and ∠BDC are neighbor angles and their sum is equal to 180°.

∠BDC is an exterior angle and ∠BDC=∠ABD+∠ADB.

Similarly, ∠ADB is an exterior angle and ∠ADB=∠CBD+∠BCD

Step-by-step explanation: