5. A gym membership costs $30 to join and $12 each month. Write and use an algebraic expression to find the cost of the gym membership for 6 months.

Final answer:

The five key elements of a two column geometric proof include: Statement, Given, Prove, Logical Arguments, and a Diagram (if applicable). These components are crucial in presenting a clear and logical proof based on mathematical principles.

Explanation:

The five key elements of a two-column geometric proof are essential for laying out the reasoning behind geometric propositions in a clear and structured way. When constructing or analyzing these proofs, the following elements should be included:

Each of these elements plays a crucial role in ensuring that the proof is valid, understandable, and follows a logical sequence based on accepted mathematical principles.

Final answer:

To calculate the total seconds a clock gains in six months, we multiply the number of minutes in six months by the gain per minute, resulting in a total of 5,184 seconds gained.

Explanation:

The question asks us to determine how many seconds a clock gains over a six-month period, assuming each month has 30 days. The clock gains 0.020 seconds per minute.

To find the total seconds gained, we need to calculate the number of minutes in six months and then multiply that by the gain per minute. There are 60 minutes in an hour, 24 hours in a day, and the problem assumes 30 days in a month. Over six months, this totals:

Now, multiply the total minutes by the gain per minute:

Therefore, the clock will gain a total of 5,184 seconds over a six-month period assuming each month has 30 days.

The correct answer is A. Crossing the street.

Explanation

Being an observer refers to observing or looking at every detail of the surrounding environment or situation. This implies when you are an observer you understand a situation by noticing and understanding every detail. In this task, individuals mainly use their sense of sight, which allows individuals to understand visually elements, although observing often involves other senses such as hearing. This is necessary in the cases we need to remember information, perform precise actions, among others.

According to this, it is necessary to be an observer when crossing a street because this task or process requires concentration and precise actions and by observe you make sure that there is no danger of being hit by a car before crossing. So, the correct answer is A. Crossing the street

Simultaneous equations can be solved if the number of unknown variables are equal to the number of equations. In the geometry question, the equations are given by the relationship between the lines and angles

Part A

m = 7°, p = 10°, t = 5°, a = 6, and s = 2°

Part B

Arranging the variables from least to greatest gives; stamp

A stamp sits at the corner of an envelope but travels around the world

Part A

From the diagram, we have;

H is a vertical angle to the 90° angle

(7·m + 3)° + 90° + (8·m - 18)° = 180° (linear angles)

∴ (7·m + 3)° + (8·m - 18)°  = 180° - 90° = 90°

(15·m - 15)° = 90°

15·m = 90°+ 15° = 105°

m = 105°/15 = 7°

m = 7°

(7·m + 3)° = (5·p + 2)° (vertically opposite angle theorem)

∴ (5·p + 2)° = (7 × 7 + 3)° = 52°

p = (52° - 2°)/5 = 10°

p = 10°

(8·m - 18)° = (11·t - 17)° (vertically opposite angle theorem)

∴ (8 × 7 - 18)° = (11·t - 17)°

38° = (11·t - 17)°

11·t = 38° + 17° = 55°

t = 55°/11 = 5°

t = 5°

CD = 5·a + 12 and DE = 9·a - 12 are congruent segments

5·a + 12 = 9·a - 12 (Definition of congruent segments)

9·a - 5·a = 12 + 12 = 24

4·a = 24

a = 24/4 = 6

a = 6

(21·s + 6° and 48° are congruent congruent angles

21·s + 6° = 48° (Definition of congruent angles)

∴ s = (48° - 6°)/21 = 2°

s = 2°

Part B

Arranging the values of the variables from least to most gives;

s = 2°

t = 5°

a = 6

m = 7°

p = 10°

Which gives;

The answer to the riddle = s, t, a, m, p = Stamp

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8 weeks the coat would cost 150 * (1-0.15+ = 127.50

9 week it would cost 127.50 *(1-0.05) = 121.13

10 week = 121.13 * (1-0.05) = 115.07

 week 11 = 115.05 * (1-0.05) = 109.32

he can buy it in week 11, because it would be less than 110 and the sore would still make a profit, because it costs more than 100

To achieve a future value of $5000 after 4 years with a 6% rate of return, Pablo would need to invest an initial amount.

To calculate the future value of an investment with a given rate of return, you can use the formula:

FV = PV * (1 + r)ⁿ

Where FV is the future value, PV is the present value, r is the rate of return expressed as a decimal, and n is the number of years.

In this case, Pablo wants to have $5000 after 4 years with a 6% rate of return. Assuming he doesn't invest any initial amount, we can substitute these values into the formula:

FV = $0 * (1 + 0.06)⁴

FV = $0 * 1.262476

FV = $0

Based on the calculation, it is not possible for Pablo to have $5000 after 4 years with a 6% rate of return if he doesn't invest any initial amount.

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Answer:

Rent of a TV for one year would be $546.

Step-by-step explanation:

A certain store is advertising this deal :

"Rent a TV for only $10.50 a week."

We have to calculate the rent for a year.

As we know there are 52 weeks in a year.

∵ Rent of a TV for one week = $10.50

∴ Rent of a TV for one year = 10.50 × 52

                                             = $546.00

Rent of a TV for one year would be $546.

Kayla will take 8 months to add $45.00 into her bank account to reach the amount of $560.00 so that she can buy her laptop.

Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.

Let's say it will take an "x" number of months to have $560.00.

Total money = fixed deposit + 45(number of months)

Given that,

Fixed deposit = $200

560 = 200 + 45x

45x = 360

x = 8

Hence "Kayla will take 8 months to add $45.00 into her bank account to reach the amount of $560.00 so that she can buy her laptop".

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Answer:

[tex]y = \frac{48}{x}[/tex]

Step-by-step explanation:

Inverse variation:

if [tex]y \propto \frac{1}{x}[/tex]

then the equation is in the form of:

[tex]y = \frac{k}{x}[/tex]                 ....[1]

where, k is the constant of variation.

As per the statement:

When x = 3, y = 16 and when x = 6, y = 8.

Substitute the value of x and y to find k.

Case 1.

When x = 3, y = 16

then;

[tex]16=\frac{k}{3}[/tex]

Multiply by 3 both sides we have;

48 = k

or

k = 48

Case 2.

When x = 6, y = 8

then;

[tex]8=\frac{k}{6}[/tex]

Multiply by 6 both sides we have;

48 = k

or

k = 48

In both cases, we get constant of variation(k) = 48

then the equation we get,

[tex]y = \frac{48}{x}[/tex]

Therefore, the inverse variation equation can be used to model this function is, [tex]y = \frac{48}{x}[/tex]

perimeter of rectangle = 20+20 +15+15 = 70 yards = 70*3 = 210 feet

210/5 = 42 seconds

 Courtney walks it in 42 seconds

the diagonal = 15^s + 20^2 = 225+400=625

sqrt(625) = 25 yards

Natalie walks 20 +15 +25 = 60 yards

60*3 = 180 feet

180 /5 =36 seconds

42 - 36 =  6 seconds faster

In the shown image of construction, first a line constructed.

Then they draw an angle.

They make an arc to represent the angle on the constructed line in step 1.

They cut the arc by measuring the angle.

They kept open same width of the compass and they cut another arc above.

They joined the points to make equal corresponding angles.

After joining points, we got a parallel line we draw in first point.

Therefore, correct option is:

Answer:

Option 1 is correct. The line created will be parallel to the given line and passing through a point not on the given line.

Step-by-step explanation:

In the given image, firstly a line is constructed.

After drawing the line, an intersecting line is drawn. Now, the measure of angle between these two lines is calculated.

After calculating the measure of angle, an angle of same measure is created at a point not on the first line.

Then, a line is drawn passing through that point with the same angle measure.

The created line will now be parallel to the first line because the corresponding angles are equal.

Therefore, option 1 is correct. The line created will be parallel to the given line and passing through a point not on the given line.

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Answer:

The correct option is 3.

Step-by-step explanation:

The table of values is

Number of days (x)   Rent (dollars) (y)

               0                         12

               1                          21

               2                         30

               3                         39

               4                         48

It is given that Thomas wants to rent a lawn mower. he has to pay a fixed base cost plus a daily rate for renting the lawn mower.  It means the relationship between x and y is linear.

The slope intercept form of a line is

[tex]y=mx+b[/tex]              .... (1)

where, m is slope and b is y-intercept.

Slope formula:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

The line passes through (0,12) and (1,21). The slope of the line is

[tex]m=\frac{21-12}{1-0}=9[/tex]

The slope of the line is 9. From the given table it is clear that the y-intercept or initial value of the function is 12.

Substitute m=9 and b=12 in equation (1).

[tex]y=9x+12[/tex]

The required equation is y = 9x + 12. Therefore the correct option is 3.

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A triangle must be either obtuse, acute, or right.

An obtuse triangle can be dissected into no fewer than seven acute triangles

I saw this in my brothers math book. Hope this helps 

Answer:

Steps are explained in the explanation part and graph is attached below.

Step-by-step explanation:

We have to graph the function [tex]y=3.7^{-x}+2[/tex] using the rule for transformation.

Let us suppose the parent function is [tex]y=3.7^x[/tex]. Below, is the graph of the parent function (fig 1)

When we add some constant 'c' in the function then the graph of the parent function shifts upward by 'c' units.

Hence, when we add 2 then the parent function will get shifted upward by 2 units. The graph of  [tex]y=3.7^{x}+2[/tex] is attached below (fig 2)

Finally, x is replaced by -x hence the graph is reflected about y axis.

In figure 3 the graph of the given function is shown.

The graph of the function [tex]y = 3.7^{-x}+2[/tex] can be draw by using transformations. First shift the graph of [tex]y = 3.7^{x}[/tex] upwards by 2 units and then replace x by -x in the function [tex]y = 3.7^x+2[/tex] and graph of the function  [tex]y = 3.7^x+2[/tex] is reflected about y-axis.

Given :

[tex]y = 3.7^{-x}+2[/tex]

To graph the above function using transformations following steps can be use:

Step 1 - Draw the graph of exponential function [tex]y = 3.7^{x}[/tex].

Step 2 - Now, shift the graph upwards in y axis by 2 units. The resulting graph is of function [tex]y = 3.7^x+2[/tex].

Step 3 - Now, replace x with -x in the function [tex]y = 3.7^x+2[/tex] and graph of the function  [tex]y = 3.7^x+2[/tex] is reflected about y-axis. The resulting graph is of the function [tex]y = 3.7^{-x}+2[/tex] .

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We have to write 5.378 E9 in standard form.

We can write  5.378 E9 as [tex]5.378\times 10^9[/tex]

Now, in order to write it in standard form we can write [tex]10^9=1000000000[/tex]

Now, we can multiply 5.378 with 1000000000

[tex]5.378\times 1000000000\\=5378000000[/tex]

Therefore, the standard form of 5.378 E9 is given by 5378000000

The y-intercept tells us where a line intersects the y-axis and the rate of change (slope) shows how much 'y' changes for a unit change in 'x'. Given these, we can identify different line characteristics based on whether the lines have the same or different y-intercepts and slopes.

Let's start by understanding the concepts of y-intercepts and rate of change. The y-intercept describes where a line intersects the y-axis. For example, in Figure A1, the y-intercept is 9, which means the line crosses the y-axis at y=9. Similarly, the rate of change, also known as the slope, indicates how much 'y' changes for each change in 'x'. A straight line will have the same slope at all points along the line, as illustrated in the example of Figure A1 where the slope is 3.

Now, let's compare the items:
A. When the y-intercepts are the same but the rates of change are different, the lines start at the same point on the y-axis, but diverge as 'x' increases because the slope of each line is different.
B. Different y-intercepts and different rates of change means that the lines start at different points on the y-axis and vary in how steeply they climb or fall.
C. When items have the same y-intercept and the same rate of change, these lines are coincident; they will overlap perfectly because they have the same starting point on the y-axis and ascend or descend at the same rate.
D. When the rates of change are the same but the y-intercepts are different, the lines are parallel. They rise or fall at the same rate, but they do not start at the same point on the y-axis.

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The y-intercepts are the same, but the rates of change are different. The option (C) is correct.

To compare the y-intercepts and the rates of change of the given linear equation and the data in the table, we need to first understand each component:

  - The slope (rate of change) is the coefficient of [tex]\( x \),[/tex] which is [tex]\( 2 \).[/tex]

  - The y-intercept is the constant term, which is [tex]\( -4 \).[/tex]

Let's find the rate of change (slope) and the y-intercept for the data in the table.

The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\) and \((x_2, y_2)\)[/tex] is calculated as:

[tex]\[m = \frac{y_2 - y_1}{x_2 - x_1}\][/tex]

We'll calculate the slope using the points [tex]\((x, y)\)[/tex] from the table:

Using the points [tex]\((-4, -6)\) and \((-2, -5)\):[/tex]

[tex]\[m = \frac{-5 - (-6)}{-2 - (-4)} = \frac{-5 + 6}{-2 + 4} = \frac{1}{2}\][/tex]

Using the points [tex]\((-2, -5)\) and \((0, -4)\):[/tex]

[tex]\[m = \frac{-4 - (-5)}{0 - (-2)} = \frac{-4 + 5}{0 + 2} = \frac{1}{2}\][/tex]

We can see the rate of change is consistently [tex]\( \frac{1}{2} \).[/tex]

We can use the slope-intercept form [tex]\( y = mx + b \)[/tex]. We know [tex]\( m = \frac{1}{2} \)[/tex] and we can use any point from the table to find [tex]\( b \).[/tex]

Using the point [tex]\((0, -4)\):[/tex]

[tex]\[y = \frac{1}{2}x + b \\-4 = \frac{1}{2}(0) + b \\-4 = b\][/tex]

So the y-intercept [tex]\( b \)[/tex] is [tex]\( -4 \).[/tex]

Rate of change (slope):

 - Equation [tex]\( y = 2x - 4 \)[/tex] has a slope of [tex]\( 2 \).[/tex]

 - Table data has a slope of [tex]\( \frac{1}{2} \).[/tex]

Y-intercept:

Both the equation and the table data have a y-intercept of [tex]\( -4 \).[/tex]

Therefore, the correct comparison is (C) The y-intercepts are the same, but the rates of change are different.

The complete question is:

Compare the y-intercepts and the rates of change of the following items.

(A) The items have different y-intercepts and different rates of change.

(B) The rates of change are the same, but the y-intercepts are different.

(C) The y-intercepts are the same, but the rates of change are different.

(D) The items have the same y-intercept and the same rate of change.

The gradient of the line y=4x+7 is 4 and the y-intercept is 7. This line rises by 4 units vertically for every 1 unit it moves horizontally, and it intersects the y-axis at the point (0,7).

In the given line equation y=4x+7, the coefficient of x is the gradient (or slope) of the line and the constant term is the y-intercept. So, in this case, the gradient of the line is 4 and the y-intercept is 7. This means the line rises by 4 units vertically for every 1 unit it moves horizontally, and it intersects the y-axis at the point (0,7).

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