Denis and Dasha solve this problem in two different ways. The perimeter of a rectangle is 54 centimeters. Its length is 6 centimeters. What is its width? Use the drop-down menus to complete the sentences below.

The question involves a mathematical trick to determine the location of a penny based on the parity of a sum and a take-home experiment on the balancing of pennies and torque. Additionally, it touches upon probability in the context of half-life and independent probabilities.

In the trick described, you are presenting a simple mathematical puzzle to determine the location of a penny based on the parity of the sum of two numbers. This is an application of basic arithmetic operations and the concept of even and odd numbers. However, for the take-home experiment involving balancing pennies on a ruler, this is related to physics and moment of force (torque). In this experiment, to balance one penny placed 8 cm away from the pivot, you would need to find a position where the total torque caused by the two or three pennies counterbalances the torque caused by the single penny. This involves understanding the principles of levers and torque.

the half-life coin activity mentioned in the question is an example of using probability to model physical phenomena, specifically radioactive decay. It explores independent probabilities when flipping coins multiple times.

These experiments and demonstrations help to illustrate various scientific and mathematical concepts using coins, which are familiar objects, making them more approachable and relatable to students.

Answer:

You bring home $14.25 with you.

Step-by-step explanation:

First we add the money spent in movie tickets and snacks.

Your ticket = $7.25

Brother's ticket = $4.25

Sister's ticket = $4.25

Soda and popcorn = $20.00

Total = 7.25 + 4.25 + 4.25 + 20.00 = $35.75

Your parents gave you = $50.00

You spent = $35.75

Balance = 50.00 - 35.75 = $14.25

You bring home $14.25 with you.

Final answer:

The direct variation equation relating Lee's pay (x) to the hours worked (y) is y = 7.60x. Lee's pay for working 25 hours in a week is $190.

Explanation:

Let's define the direct variation equation:

y = kx

We can substitute the information given in the question:
When x = 6.5, y = 49.40
49.40 = 6.5k
k = 49.40 / 6.5
k = 7.60

Therefore, the direct variation equation relating Lee's pay (x) to the hours worked (y) is y = 7.60x.

To find Lee's pay if she works 25 hours, we can substitute x = 25 into the equation:
y = 7.60(25)
y = 190

Lee's pay for working 25 hours in a week is $190.

The property that justifies the step is the Commutative Property of Addition, which states that the order of addition does not affect the result.

The property that justifies the illustrated step in solving the equation 3x + 4 - 2x + 5 = 6x + 2 as 3x - 2x + 4 + 5 = 6x + 2 is the Commutative Property of Addition.

This property states that the order in which numbers are added does not affect the result. So, we can rearrange the terms in any order without changing their sum.

In this step, we are rearranging the terms -2x and 5 so that like terms, namely 3x and -2x, and 4 and 5, are grouped together.

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After stretching for 8 minutes, Leticia had 30 minutes left for running. Given that she takes 6 minutes to complete one round, she completed 5 rounds in the remaining time.

The question asks how many times Leticia jogged around her block after stretching for 8 minutes within her total exercise time of 38 minutes. Let's deduct the time Leticia spent stretching from the total time to find out how much time she spent running. That is 38 - 8 = 30 minutes. Since we know that running around the block once takes Leticia 6 minutes, we then divide her running time by this to discover how many laps she was able to make.

We do this by 30 / 6 = 5. Therefore, Leticia jogged around her block 5 times.

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Probability measures to determine the possibility of an event. In this case, the events are cars with no 4 wheels and cars with third row seats.

The probability that a randomly selected car with no 4-wheel drive has third row seats is 0.3

Let:

[tex]A \to[/tex] A car with no 4-wheel drive

[tex]B \to[/tex] A car that has third row seats

The probability that a randomly selected car with no 4-wheel drive has third row seats is represented as:

[tex]P(B|A)= \frac{n(A\ n\ B)}{n(A)}[/tex]

From the 2-way table, we have the following parameters:

[tex]n(A\ n\ B) = 12[/tex]

[tex]n(A) = 40[/tex]

So, the probability is:

[tex]P(B|A)= \frac{n(A\ n\ B)}{n(A)}[/tex]

[tex]P(B|A)= \frac{12}{40}[/tex]

[tex]P(B|A)= 0.3[/tex]

Hence, the probability that a randomly selected car with no 4-wheel drive has third row seats is 0.3

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Answer: 28.1ft^2 so it’s C

Answer: C

Step-by-step explanation:

Answer:

#1) d. ΔJKL is not a right triangle because no two of its sides are perpendicular; #2) -1/3, 3, -7, is, two of these slopes have a product of -1; #3) a. Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length; #4) 1, -1/6, 1, -2/5, is not, only one pair of opposite sides is parallel; #5) c. Quadrilateral PQRS is not a rectangle because it has only one right angle.

Step-by-step explanation:

#1) The slope of any line segment is found using the formula

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

For JK, this gives us (1-1)/(-5-0) = 0/-5 = 0.  For KL this gives us (1--5)/(0-2) = 6/-2 = -3.  For LJ this gives us (-5-1)/(2--5) = -6/7.  None of these slopes are negative reciprocals, so none of the angles are right angles and this is not a right triangle.

#2) The slope of JK is (2-1)/(0-3) = 1/-3 = -1/3.  The slope of KL is (1--5)/(3-1) = 6/2 = 3.  The slope of LJ is (2--5)/(0-1) = 7/-1 = -7.  Two of these slopes have a product of -1, 3 and -1/3.  This means they are negative reciprocals so this has a right angle; this means JKL is a right triangle.

#3) The slope of DE is (5-4)/(-2-2) = 1/-4 = -1/4.  The slope of EF is (4-0)/(2-0) = 4/2 = 2.  The slope of FG is (0-1)/(0--4) = -1/4.  The slope  of GD is (1-5)/(-4--2) = -4/-2 = 2.  Opposite sides have the same slope so they are parallel.

Next we use the distance formula to find the length of each side:

[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Using our points, the length of DE is

[tex]\sqrt{(5-4)^2+(-2-2)^2}=\sqrt{1^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of EF is

[tex]d=\sqrt{(4-0)^2+(2-0)^2}=\sqrt{4^2+2^2}=\sqrt{16+4}=\sqrt{20}[/tex]

The length of FG is

[tex]d=\sqrt{(0-1)^2+(0--4)^2}=\sqrt{(-1)^2+(4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of GD is

[tex]d=\sqrt{(1-5)^2+(-4--2)^2}=\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}[/tex]

Opposite sides have the same length and are parallel, so this is a parallelogram.

#4) The slope of AB is (-1-2)/(-4--1) = -3/-3 = 1.  The slope of BC is (2-1)/(-1-5) = 1/-6 = -1/6.  The slope of CD is (1--3)/(5-1) = 4/4 = 1.  The slope of DA is (-3--1)/(1--4) = -2/5.  Only one pair of opposite sides is parallel, so this is not a parallelogram.

#5) The slope of PQ is (2-4)/(-4-3) = -2/-7 = 2/7.  The slope of QR is (4-0)/(3-5) = 4/-2 = -2.  The slope of RS is (0--2)/(5--3) = 2/8 = 1/4.  The slope of SP is (-2-2)/(-3--4) = -4/1 = -4.  Only one pair of sides has slopes that are negative reciprocals; this means this figure only has 1 right angle, so it is not a rectangle.

The answers provided for ΔJKL, ΔJKL as a right triangle, and quadrilateral ABCD are correct. For quadrilateral DEFG, the answer may be correct depending on side lengths, and for quadrilateral PQRS, slopes must be calculated to determine if it's a rectangle.

To determine whether the given shapes are right triangles, rhombuses, rectangles, or parallelograms, we use properties such as the slopes of lines to check for perpendicularity, parallelism, and equal lengths.

For ΔJKL with vertices J(-5, -1), K(0, 1), L(2, -5), you calculated it's not a right triangle because none of the slopes of the sides are negative reciprocals of each other (indicating perpendicular sides). This is correct; no slopes have a product of -1, so answer (d) is correct.

In the second case, ΔJKL with vertices J(0, 2), K(3, 1), L(1, -5) has slopes -1/3, 3, and -7 for sides JK, KL, and JL respectively. Since -1/3 and 3 are negative reciprocals, ΔJKL is a right triangle due to the perpendicular sides JK and KL. So, the filled answers are correct.

Regarding quadrilateral DEFG, you might want to reconsider option (b), as you cannot determine whether it's a rhombus solely based on one pair of opposite sides being parallel. Instead, check if all sides have the same length using the distance formula, and if the opposite sides are parallel by comparing slopes. Since you didn't provide the side lengths, I will only address the slopes here, which you've noted. Answer (b) can be correct if the side lengths are not equal, but if they are, it should be (c).

For quadrilateral ABCD, your assessment of the slopes leads to the conclusion that it is not a parallelogram, which is correct as only one pair of opposite sides (AB and CD having a slope of 1) is parallel. The filled answers are appropriate.

Last, for quadrilateral PQRS, you must calculate the slopes of the sides to determine if the sides are perpendicular to each other by checking if the products of the corresponding slopes are -1. If the slopes of adjacent sides are negative reciprocals of each other, then PQRS has right angles, and it could be a rectangle. Calculate the slopes, and if none are negative reciprocals, then answer (d) is correct stating that PQRS is not a rectangle because it has no right angles.

Answer:

The correct option is A.

Step-by-step explanation:

Let number of sold wristbands and headbands be x and y respectively.

It is given that the jack earns $2 for each wristband and $3 for each headband. So, total revenue is

[tex]T=2x+3y[/tex]

He wants to earn at least $50.

[tex]2x+3y\geq 50[/tex]

He needs to sell at least 5 wristbands.

[tex]x\geq 5[/tex]

The number of sold wristbands and headbands can not be negative. so,

[tex]x\geq 0,y\geq 0[/tex]

The system of inequalities is

[tex]2x+3y\geq 50[/tex]

[tex]x\geq 5[/tex]

[tex]x\geq 0,y\geq 0[/tex]

Therefore, correct option is A.

Answer:

Landon makes $10.5 per hour at babysitting and $7.75 per hour at water park.

Step-by-step explanation:

Let the amount made per hour in dollars for babysitting be = x

Let the amount made per hour in dollars at water park be = y

As per the condition, the equations form:

[tex]3x+10y=109[/tex]     .......(1)

[tex]8x+12y=177[/tex]     ......(2)

Multiplying (1) by 8 and (2) by 3, and subtracting (2) from (1), we get

[tex]44y=341[/tex]

=> y = 7.75

And [tex]3x+10(7.75)=109[/tex]

=> [tex]3x+77.5=109[/tex]

=> [tex]3x=109-77.5[/tex]

=> [tex]3x=31.5[/tex]

x = 10.5

Therefore, Landon makes $10.5 per hour at babysitting and $7.75 per hour at water park.

The answer is x=1. Just did the assignment

The equation is a complex exponentiation problem that can be solved by simplifying and rearranging the equation. By plugging in the given potential solutions, it can be determined that x = -1 is the correct answer.

This equation is an exponential equation. You can start solving by simplifying the bases, as both bases are actually the 2, but to different powers (512 = 2^9 and 1/64 = 2^-6). The equation can be rewritten as (2^9)^x - 2 = (2^-18)x - 512. Simplifying this gives 2^(9x) - 2 = 2^(-18x) - 512. From the options given (x = -1, 0, or 1), we can plug these into the simplified equation to see which one gives a valid solution. Plugging these values in reveals that x = –1 is the valid solution to the equation. None of the other choices work out correctly.

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If there are 812 students in your school and there are 36 more girls than boys. Then there are 414 girls in the school.

In this problem, let's denote the number of boys as B and the number of girls as G. We are given two pieces of information: there are 812 students in total, and there are 36 more girls than boys.

So, we can set up two equations based on these statements:

1. Total number of students: (B + G = 812)

2. The number of girls is 36 more than boys: (G = B + 36)

Now, we can substitute the expression for G from the second equation into the first equation:

[B + (B + 36) = 812]

Combine like terms:

[2B + 36 = 812]

Subtract 36 from both sides:

[2B = 776]

Now, divide by 2 to solve for B:

[B = 388]

Now that we know the number of boys, we can find the number of girls using the second equation:

[G = 388 + 36]

[G = 424]

So, there are 388 boys and 424 girls in the school. To find the number of girls, we add the number of boys (388) to the given difference (36):

[424 = 388 + 36]

Therefore, the final answer is that there are 424 girls in the school.