Ethan and Calvin meet daily in the elevator. One morning they found that if they multiply their current ages, they get 238. If they did the same after four years, this product would be 378. Determine the sum of the current ages of Ethan and Calvin.

The value of x with the answers in simplest radical form is 18.

We are given that;

The right triangles with dimension

Now,

The Pythagoras theorem states that the square of the longest side must be equal to the sum of the square of the other two sides in a right-angle triangle.

|AC|^2 = |AB|^2 + |BC|^2

1. 14^2+14^2=x^2

196+196=x^2

x^2=392

x=19.7

2. x^2+X^2=12^2

2x^2=144

x^2=72

x=8.48

3. 9[tex](\sqrt{2} )^2[/tex]+9[tex]\sqrt{2})^2[/tex]=x^2

162+162=x^2

324=x^2

x=18

Therefore, by pythagoras theorem the answer will be 18.

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The two other positive angles of rotation that take Point A to Point B on the unit circle is (5π/6) and (5π/6) + 2π .

Given data ,

To find two other positive angles of rotation that take Point A to Point B, we need to consider the angle values that yield the same coordinates as (1, 0) after rotating counterclockwise.

The position of Point A is (1, 0) on the unit circle.

Now, let's find the coordinates of Point B after rotating (7π/6) radians counterclockwise.

To rotate counterclockwise by (7π/6) radians, we can subtract (7π/6) from the angle of Point A. So, the angle for Point B would be:

Angle of Point B = 0 - (7π/6) = - (7π/6)

Now , for positive angles of rotation, we can add multiples of 2π to the angle of Point B while keeping the same coordinates. Adding 2π to the angle gives us:

Angle of Point B = - (7π/6) + 2π = (5π/6)

Hence , two other positive angles of rotation that take Point A to Point B are (5π/6) and (5π/6) + 2π. Both of these angles yield the same coordinates as Point B, which is (1, 0) on the unit circle.

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Answer: To rewrite the fractions 2/3 and 4/15 with the least common denominator (LCD), we need to find the smallest multiple that both denominators, 3 and 15, divide into evenly.

The prime factorization of 3 is 3, and the prime factorization of 15 is 3 * 5.

To find the LCD, we take the highest power of each prime factor that appears in either denominator. In this case, the highest power of 3 is 3, and the highest power of 5 is 5.

The LCD is the product of these highest powers: LCD = 3 * 5 = 15.

Now, we can rewrite the fractions with the least common denominator:

2/3 = (2/3) * (5/5) = 10/15

4/15 = (4/15) * (1/1) = 4/15

Therefore, the fractions 2/3 and 4/15 can be rewritten with the least common denominator as 10/15 and 4/15, respectively.

Step-by-step explanation:

Answer:

The median is the middle value in the set. Since there are 5 values in this set, the middle value is the third value, which is 6.

Therefore, the median of the set {2, 4, 6, 8, 10} is 6.

Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).

In order to find the missing expression in Step 1, let's analyze the given steps and the final expression.

Step 1: ?

Step 2: -9 + 2 + 3

Step 3: -7 + 3

To find the missing expression in Step 1, we need to work backwards from Step 3 to Step 1.

In Step 3, the expression "-7 + 3" gives us a result of -4.

In Step 2, the expression "-9 + 2 + 3" gives us a result of -4.

So, the missing expression in Step 1 should also evaluate to -4 when performed correctly.

Let's check the available options:

[-10 + -1 + 2] + (2 + 3) = -11 + 2 + 5 = -4

[-8 - 1] + (2 + 3) = -9 + 5 = -4

[-10 + 1] + (2 + 3) = -9 + 5 = -4

[8 + 1] + (2 + 3) = 9 + 5 = 14

Out of the given options, only option 2, [-8 - 1] + (2 + 3), correctly evaluates to -4. Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).

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Consecutive Interior Angles.

They are on the same side of a line that cuts through two other lines.

The probability that the first ball chosen is black and the second ball chosen is red is 24.24%

First, we shall obtain the total balls present in the box. Details below:

Total ball = 4 + 8

Total ball = 12 balls

Next, we shall obtain the probability of chosen a black ball first. Details below:

P(B) = n(B) / n(T)

P(B) = 8 / 12

Next, we shall obtain the probability of chosen red as the 2nd ball. Details below:

P(R | B)= n(R) / n(T)

P(R | B) = 4 / 11

Finally, we shall obtain the probability that the first ball chosen is black and the second ball chosen is red. Details below:

P(B ∩ R) = [P(B) × P(R | B)] × 100

P(B ∩ R) = [(8 / 12) × (3 / 11)] × 100

P(B ∩ R) = 0.2424 × 100

P(B ∩ R) = 24.24%

Thus, the probability that the first ball chosen is black and the second ball chosen is red is 24.24%

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Although the rectangles have the same base lengths, the heights and areas are not directly proportional in this case.

Let's denote the base length of both rectangles as 'b'. If one rectangle has a height that is twice the height of the other rectangle, we can denote the heights as 'h' and '2h', respectively.

The area of a rectangle is calculated by multiplying the base length by the height. Therefore, the area of the first rectangle with height 'h' would be A₁ = b * h, and the area of the second rectangle with height '2h' would be A₂ = b * (2h) = 2b * h.

Comparing the two areas, we have A₁ = b * h and A₂ = 2b * h. It is evident that the areas are not proportional because the area of the second rectangle is twice the area of the first rectangle.

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Answer:

[tex]a_{n}[/tex] = - 7n + 47

Step-by-step explanation:

there is a common difference between consecutive terms , that is

33 - 40 = 26 - 33 = - 7

this indicates the sequence is arithmetic with nth term

[tex]a_{n}[/tex] = a₁ + d(n - 1)

where a₁ is the first term and d the common difference

here a₁ = 40 and d = - 7 , then

[tex]a_{n}[/tex] = 40 - 7(n - 1) = 40 - 7n + 7 = - 7n + 47

The calculated age of the sample is 2865 years. Since this is less than 5,000 years, the scientist's hypothesis that the tree bark is over 5,000 years old is incorrect.

To determine if the scientist's hypothesis is correct, we can use the concept of half-life and the given information.

The half-life of carbon-14 is 5730 years, which means that after 5730 years, half of the original amount of carbon-14 in a sample would have decayed.

In this case, the fossilized tree bark is thought to be over 5,000 years old. If the sample contains 75% of the original amount of carbon-14, it means that 25% of the carbon-14 has decayed.

Let's calculate the number of half-lives that have passed:

(100% - 75%) / 50% = 0.25 / 0.5 = 0.5

So, 0.5 half-lives have passed.

Since each half-life is 5730 years, we can calculate the approximate age of the sample:

0.5 half-lives × 5730 years per half-life = 2865 years

The calculated age of the sample is 2865 years. Since this is less than 5,000 years, the scientist's hypothesis that the tree bark is over 5,000 years old is incorrect.

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If we know that <C is congruent to <E, we can say that: (a). the two triangles are similar by SAS.

from the question, we have the following parameters that can be used in our computation:

The triangles (see attachment)

Two triangles are similar when the ratio of their corresponding sides are equal and their corresponding angles are congruent.

So, we have

Ratio = BC/AC = 6/3 = 2.

Ratio = FE/DE = 4/2 = 2.

Since we know that <C is congruent to <E, we can say that the two triangles are similar by side, angle, side (SAS) criterion.

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The volume of the given figure is 750 in.

Given

Base =30in

Height = 8in

Side = 15

Side = 17

Required to calculate the volume of the given figure

Volume = Base x Height + 2 side x side (there are two rectangle)

             = 30 x 8 + 2 x 15 x 17

              = 750

The region that a 2D shape covers in an area versus the space a 3D shape occupies is the main distinction between area and volume. Geometry is filled with a wide variety of shapes and sizes.

A mass is a solid body or an assortment of distinguishable visual components that together make a solid form, such as lines, colors, textures, etc. Length, width, and depth make up the three dimensions of volume.

Thus, the volume of the given figure is 750.

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The  names of the siblings from shortest to longest time are:

Rafael= 75min = 1 hour and 15 minutes.

Leanne= 1 hour and 25 minutes

Ray= 1 3/4 =  1 hour and 45 minutes.

To be able to compare 1 hour and 25 minutes with  1 3/4 (1.75 hours), one need convert the minutes to hours. 1 hour is equal to 60 minutes, so 1 hour and 25 minutes is equivalent to 1.42 hours.

1.42 hours is smaller than 1.75 hours. hence Ray has the longest time.

So according to the given information, note that:

Hence, the siblings' names listed from shortest to longest time are Rafael, Leanne, and Ray.

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The velocity represented by vector AC of Michelle's canoe is approximately option C. 11.2 miles per hour.

To determine the velocity, represented by vector AC, of Michelle's canoe, use the concept of vector addition.

Break down the motion into its components:

1. Paddling due east at 10 miles per hour: This motion can be represented by a vector pointing in the positive x-direction with a magnitude of 10 mph.

2. Wind blowing due south at 5 miles per hour: This motion can be represented by a vector pointing in the negative y-direction with a magnitude of 5 mph.

To find the resultant velocity, add these two vectors. Using the Pythagorean theorem, calculate the magnitude of the resultant velocity:

AC² = AB² + BC²

AB (paddling east) = 10 mph

BC (wind blowing south) = -5 mph (negative sign due to opposite direction)

AC² = (10 mph)² + (-5 mph)²

AC² = 100 mph² + 25 mph²

AC² = 125 mph²

Taking the square root of both sides:

AC ≈ √(125) mph

AC ≈ 11.2 mph (rounded to the nearest tenth)

Therefore, the velocity represented by vector AC of Michelle's canoe is approximately 11.2 miles per hour.

The correct answer is option C: 11.2 miles per hour.

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Using the slope-intercept form, the equation of the lines are

a. 2y = x + 9

b. 3y = x + 1

c. y = x - 6

d. y = x + 10

e. y = -1/2x + 1

In the given question, we have the coordinate of a point and the slope of the line.

To determine the equation of the straight line, we have to use the formula of slope-intercept which is given as; y = mx + c.

Plugging the values into the formula.

a. (3, 6) and slope = 1/2

equation of line = y = mx + c = 6 = 1/2(3) + c

6 = 3/2 + c

c = 9/2

The equation of line is y = 1/2x + 9/2 ; 2y = x + 9

b. The point is (2,1) and slope is 1/3

Equation of line is;

y = mx + c

1 = 1/3(2) + c

c = 1 - 2/3

c = 1/3

Equation of line is y = 1/3x + 1/3; 3y = x + 1

c. The point is (4, -2) and slope is 1

y = mx + c

-2 = 1(4) + c

c = -2 - 4 = -6

y = x - 6

d. The point is (-2, 8) and slope is 1

y = mx + c

8 = 1(-2) + c

c = 10

y = x + 10

e. The point is (-4, 3) and slope is -1/2

y = mx + c

3 = -1/2(-4) + c

3 = 2 + c

c = 1

y = -1/2x + 1

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Answer:

caculations below

Step-by-step explanation:

A conveyor belt carries supplies from the first floor to the second floor which is 23 feet higher. The belt makes a 60-degree angle with the ground.

To calculate the distance traveled by the supplies from one end of the conveyor belt to the other, we can use the trigonometric function of sine.

sin(60) = opposite / hypotenuse

Here, the opposite side is the height difference between the two floors, which is 23 feet.

sin(60) = 23 / hypotenuse

To find the hypotenuse, we can rearrange the above equation:

hypotenuse = 23 / sin(60)

Using a calculator, we get:

hypotenuse = 26.5 feet

Therefore, the supplies travel approximately 26.5 feet from one end of the conveyor belt to the other.

Rounding the answer to the nearest foot, we get:

The supplies travel approximately 27 feet from one end of the conveyor belt to the other.

The voltage drop across resistor R₃ is determined as 237.9 V.

The total current flowing in the circuit is calculated by applying Ohm's law as shown below.

V = IR

I = V / R

where;

Since the circuit is in series connection, the total resistance is calculated as follows;

R = R₁ + R₂ + R₃

R = 76 ohms + 61 ohms + 30 ohms

R = 167 ohms

The current flowing in each circuit component is calculated as;

I = V/R

I = 1325 V / 167Ω

I = 7.93 A

The voltage drop across R₃ is calculated as follows;

V₃ = IR₃

V₃ = 7.93 x 30

V₃ = 237.9 V

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The complete question is below:

This cirucit is in series. Determine the voltage dropped on R3, given:

ET = 1325 V, R1 = 76 Ω, R2 = 61 Ω, R3 = 30 Ω, and R4 = 30 Ω

1) The value of A ∪ B is,

A ∪ B = {<, +, =, /, >, x }

2) The value of A ∩ M is,

A ∩ B = {2}

3) The value of B ∩ C is,

B ∩ C = { }

4) The value of M - J is,

M - J = {soap}

5) The complement of set A is,

A' = (to, combat, coronavirus}​

Now, WE can simplify as;

1) We have;

A = {<, +, =, x) and B = {=, +, /, >}

Hence, We get;

The value of A ∪ B is,

A ∪ B = {<, +, =, /, >, x }

2) Given that,

A = {1, 2, 4) and M = {2, 3, 5}

Hence, The value of A ∩ M is,

A ∩ B = {2}

3) Given that;

B = {C, O, U, G, H) and C = {F, E, V, E, R)

Hence, The value of B ∩ C is,

B ∩ C = { }

4) Given that;

M = {soap, alcohol, sanitizer, mask) and J = {alcohol, sanitizer, mask, gloves}

Hence, The value of M - J is,

M - J = {soap}

5) Given that;

U = {Frontliners, sacrifice, their, lives, to, combat, coronavirus) and A = {Frontliners, sacrifice, their, lives)

Hence The complement of set A is,

A' = U - A

A' = (to, combat, coronavirus}​

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The relative frequency that a randomly selected student's favorite exhibit was either dinosaurs, or planets is 0.7.

To find the relative frequency, we will divide the number of students who chose dinosaurs or planets by the total number of students surveyed.

The total number of students surveyed is 12 + 25 + 17 + 6 = 60.

The number of students who chose dinosaurs or planets is 25 + 17 = 42.

Therefore, the relative frequency will then be 42 / 60 = 0.7.

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The thermal energy of the first rock is twice the Thermal energy of the second rock.

The thermal energy of an object is directly proportional to its mass. Therefore, if the first rock has twice the mass of the second rock, it will also have twice the thermal energy.

Thermal energy is a measure of the internal energy of an object, which includes the kinetic energy of its particles. In this case, since both rocks are made of the same material and are at the same temperature, we can assume that their particles have the same average kinetic energy per particle.

The thermal energy of an object can be calculated using the formula:

Thermal energy = Mass * Specific heat capacity * Change in temperature

Since the specific heat capacity and the change in temperature are the same for both rocks (as they are made of the same material and are at the same temperature), we can simplify the comparison of their thermal energies based on mass alone.

Let's denote the mass of the second rock as "m" (arbitrary unit). Since the first rock has twice the mass, its mass can be represented as "2m".

Therefore, the thermal energy of the first rock is:

Thermal energy of the first rock = (2m) * Specific heat capacity * Change in temperature

And the thermal energy of the second rock is:

Thermal energy of the second rock = m * Specific heat capacity * Change in temperature

Comparing the thermal energies:

Thermal energy of the first rock / Thermal energy of the second rock = (2m) * Specific heat capacity * Change in temperature / (m) * Specific heat capacity * Change in temperature

The specific heat capacity and the change in temperature cancel out, leaving us with:

Thermal energy of the first rock / Thermal energy of the second rock = 2m / m = 2

Therefore, the thermal energy of the first rock is twice the thermal energy of the second rock.

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Length of JN = 10

x = 6

Given ,

S is the center of the circle.

JK = 20

LM = 3x + 2

SN = 12

SP = 12

Now ,

SN and SP are perpendicular to the chords JK and LM respectively .

Perpendiculars drawn from the center of circle to the chords bisect chords into two equal halves .

Thus,

JN = JK/2

JN = 10

Now join SJ,

In ΔSJN ,

Apply pythagoras theorem,

SN² + NJ² = SJ²

12² + 10² = SJ²

SJ = 14.52

SJ =Radius of the circle .

Now,

LP = LM/2

LP = 1.5x + 1

Now join  SL,

In ΔSLP

SP² + PL² = SL²

SL = SJ (radius of circle)

So,

12² + (1.5x + 1)² = 244

x = 6

Hence the value of x is 6 and JN is 10 .

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Answer:

See below for answers and explanations

Step-by-step explanation:

[tex]\displaystyle \lim_{x\rightarrow1^{-}}f(x) = 4-(1)-(1)^2=4-1-1=3-1=2\\\\\lim_{x\rightarrow1^{+}}f(x) = 2(1)-1=2-1=1\\\\\lim_{x\rightarrow1}f(x) = DNE[/tex]

Note that [tex]\displaystyle \lim_{x\rightarrow1^-}f(x)[/tex] represents the left-side limit (so the limit of f(x) as x approaches 1 from the left), and [tex]\displaystyle \lim_{x\rightarrow1^+}f(x)[/tex] represents the right-side limit (so the limit of f(x) as x approaches 1 from the right)

Because [tex]\displaystyle \lim_{x\rightarrow1^-}f(x)\neq\displaystyle \lim_{x\rightarrow1^+}f(x)[/tex], then [tex]\displaystyle \lim_{x\rightarrow1}f(x)=DNE[/tex]

I hope this helped!

Answer:

6

Step-by-step explanation:

The correct answer is C. 6.

If we substitute 6 for x in the equation, we get 4(8-6) = 4(2) = 8.

This is the only value of x that makes the equation true.

Answer:

The remaining is 75%, the length of the stick would be 1.2

Step-by-step explanation:

According to the information given,

We know that Brian cut off 25% of a stick which was 1.6 meters long

and 1.6 is the 100% of the stick:

It is fairly easy to calculate this, subtract 25 from 100 ( 100 - 25 ), which is equal to 75.

Hence, the answer is 75% and 1.2 for the remaining length of the stick

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The area of smaller triangle is 59 square feet.

The area of the larger triangle is 105 square feet.

And the one side of the larger triangle and the smaller traingle is 16 and 12 feet respectively.

Suppose the height of thesmaller triangle be x feet and the height of the larger triangle be y feet.

Since, the corresponding edges of similar triangles are proportional.

Therefore, y/x=16/12

y=4/3x

Also, the area of larger triangle is 105 square feet.

1/2×4/3x×16=105

The above equation can be written as

1/2×4/3x×4/3×12=105

1/2×x×12=59

Area of smaller triangle  = 59 square feet

Hence, the area of smaller triangle is 59 square feet.

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Answer:

Step-by-step explanation:

Hello!

We want to know the area of the original circle - the now circle.

The original circle's diameter is 110 meters.

To figure out the area you use this formula: [tex]r^2*pi[/tex]

The diameter is 107 meters, so the equation is [tex]55^2[/tex][tex]*\pi[/tex].

The area is about 9503.3 meters.

The next area is the "now circle".

The now circle's diameter is 107 meters.

When you use this formula, you end up with  [tex]53.5^2*\pi[/tex].

The area is about 8892 meters.

So it's just basic subtraction!

I'll let you figure out the rest.

:)

The box that would be able to hold all the cubes if Robin's teacher had 200 cubes is given as follows:

Box B.

The volume of a rectangular prism, with dimensions defined as length, width and height, is given by the multiplication of these three defined dimensions, according to the equation presented as follows:

Volume = length x width x height.

Hence the volumes for each box in this problem are given as follows:

As 216 > 144, Box B is the box that could hold all of the 200 cubes.

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52.8 square inches of paper to cover the entire Pyramid.

The amount of paper needed to cover the entire pyramid, we need to determine the total surface area of the pyramid.

The pyramid consists of a square base and four triangular faces.

1. Base Surface Area:

The surface area of the square base can be calculated by finding the area of a square with sides measuring 4.4 inches. The formula for the area of a square is side length squared:

Base Surface Area = (4.4 in)² = 19.36 in²

2. Triangular Faces Surface Area:

Each triangular face of the pyramid has a base of 4.4 inches and a height of 3.8 inches. The formula for the area of a triangle is 1/2 times the base times the height:

Triangular Face Surface Area = 1/2 * (4.4 in) * (3.8 in) = 8.36 in²

Since the pyramid has four triangular faces, the total surface area of the triangular faces is:

Total Triangular Faces Surface Area = 4 * (8.36 in²) = 33.44 in²

3. Total Surface Area:

To calculate the total surface area of the pyramid, we add the base surface area to the total triangular faces surface area:

Total Surface Area = Base Surface Area + Total Triangular Faces Surface Area

                         = 19.36 in² + 33.44 in²

                         = 52.8 in²

Therefore, it would take approximately 52.8 square inches of paper to cover the entire pyramid.

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