x^2-2y=5 and 4y+z=7 write z in terms of x

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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1. The function representing the population of bacteria after t minutes is P(t) = 4900× [tex]e^{(0.08(t/60)}[/tex]

2. The percentage rate of change per minute is approximately 0.13%.

The general form of an exponential growth function ⇒ P(t) = P0 ×[tex]e^{(rt)}[/tex]

P(t) is the population at time t,

P0 is the initial population,⇒4900

r is the growth rate  ⇒ 8%

t is the time.

Therefore, our function in terms of hours (t) becomes:

P(t) = 4900 × [tex]e^{(0.08t)}[/tex]

To find per minute, there are 60 minutes in an hour,

∴ t becomes  t/60

P(t) = 4900× [tex]e^{(0.08(t/60))}[/tex]

We know the rate of change per hour is 8%, so the rate of change per minute will be 8% divided by 60, which is

[tex]e^{(0.08/60)-1}[/tex] × 100 = 0.13%.

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Answer:

2

Step-by-step explanation:

Ujeli had 11 pencils and she kept 3 so 11-3 = 8

so that means that the remaining pencil were dived by 4 (8÷2) = 2

The work done in stretching the spring from 10 centimeters to 40 centimeters is 9000 joules.

To determine the work done in stretching the spring from 10 centimeters to 40 centimeters, we need to understand the relationship between force, displacement, and work.

In this case, we are given that a force of 450 newtons stretches the spring by 30 centimeters.

We can use this information to calculate the spring constant (k) using Hooke's Law, which states that the force applied to a spring is directly proportional to the displacement it undergoes.

Hooke's Law: F = kx

Where:

F is the force applied,

k is the spring constant, and

x is the displacement of the spring.

We can rearrange the equation to solve for k:

k = F / x

k = 450 N / 30 cm

k = 15 N/cm

Now that we have the spring constant, we can calculate the work done in stretching the spring from 10 cm to 40 cm using the formula for work:

Work [tex]= (1/2)k(x2^2 - x1^2)[/tex]

Where:

k is the spring constant,

x1 is the initial displacement, and

x2 is the final displacement.

Plugging in the values:

Work [tex]= (1/2)(15 N/cm)(40 cm^2 - 10 cm^2)[/tex]

Work [tex]= (1/2)(15 N/cm)(1200 cm^2)[/tex]

Work [tex]= (1/2)(15 N/cm)(1200 cm^2)[/tex]

Work = 9000 N·cm or 9000 joules (since 1 N·cm = 1 joule)

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Susan would have to repay her parents $8,454.96 at the end of 13 months to clear the outstanding balance using the declining balance method.

To calculate the outstanding balance at the end of 13 months, we need to use the declining balance method. This method involves applying the interest rate to the remaining balance after each repayment.

First, let's calculate the interest for the first 4 months. The interest for this period can be calculated using the formula: Interest = Principal × Rate × Time. In this case, the principal is $18,500, the rate is 5.00% per year (or 0.05), and the time is 4 months (or 4/12 years). Thus, the interest for the first 4 months is $18,500 × 0.05 × (4/12) = $308.33.

Next, we subtract the repayment made after 4 months ($5,300) from the outstanding balance ($18,500) and add the interest for the first 4 months ($308.33). This gives us a new outstanding balance of $18,500 - $5,300 + $308.33 = $13,508.33.

Now, let's calculate the interest for the period between 4 and 10 months. The principal remains the same ($13,508.33), the rate is still 5.00% per year (or 0.05), and the time is 6 months (or 6/12 years). Therefore, the interest for this period is $13,508.33 × 0.05 × (6/12) = $337.71.

Subtracting the repayment made after 10 months ($3,800) from the outstanding balance ($13,508.33) and adding the interest for the period gives us a new outstanding balance of $13,508.33 - $3,800 + $337.71 = $10,045.04.

Finally, to determine the last payment required to clear the outstanding balance at the end of 13 months, we subtract the new outstanding balance ($10,045.04) from the principal ($18,500). The last payment is therefore $18,500 - $10,045.04 = $8,454.96.

Therefore, Susan would have to repay her parents $8,454.96 at the end of 13 months to clear the outstanding balance using the declining balance method.

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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:

Aprox: 465.4 ms/2

Step-by-step explanation:

To find the magnitude of the acceleration the engine must produce, we'll break down the missile's motion into its horizontal and vertical components.

Given:

Initial velocity (V₀) = 1810 m/s

Launch angle (θ) = 20.0°

Target distance (d) = 19,500 m

Target direction (φ) = 32.0°

Time of flight (t) = 9.20 s

First, let's find the horizontal and vertical components of the missile's initial velocity (V₀x and V₀y).

V₀x = V₀ * cos(θ)

V₀y = V₀ * sin(θ)

V₀x = 1810 m/s * cos(20.0°) ≈ 1712.4 m/s

V₀y = 1810 m/s * sin(20.0°) ≈ 618.8 m/s

Now, let's find the horizontal and vertical displacements (dx and dy) of the missile using the time of flight (t) and the target distance (d).

dx = d * cos(φ)

dy = d * sin(φ)

dx = 19,500 m * cos(32.0°) ≈ 16,402.8 m

dy = 19,500 m * sin(32.0°) ≈ 10,236.8 m

Next, let's calculate the acceleration needed in the x-direction (ax) and y-direction (ay) to cover the horizontal and vertical displacements in the given time.

ax = (2 * dx) / t²

ay = (2 * dy) / t²

ax = (2 * 16,402.8 m) / (9.20 s)² ≈ 391.7 m/s²

ay = (2 * 10,236.8 m) / (9.20 s)² ≈ 257.7 m/s²

Finally, to find the magnitude of the acceleration, we can use the Pythagorean theorem:

acceleration magnitude (a) = √(ax² + ay²)

a = √(391.7 m/s²)² + (257.7 m/s²)² ≈ 465.4 m/s²

Therefore, the magnitude of the acceleration that the engine must produce is approximately 465.4 m/s².

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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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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The surface area of each solid is listed below:

Case A: A = 584.988 ft²

Case B: A = 320π in²

In this problem we must determine the surface area of each of two solids, that is, the sum of the areas of all faces, whose area formulas are introduced below:

Square

A = l²

Where l is the side length.

Triangle

A = 0.5 · w · h

Where:

Circle

A = π · r²

Where r is the radius.

Lateral side of a cylinder

A = 2π · r · h

Where h is the height of the cylinder.

Now we proceed to determine the surface area of each solid:

Case A:

A = (14 ft)² + 4 · 0.5 · (14 ft) · √[(7 ft)² + (12 ft)²]

A = 584.988 ft²

Case B:

A = 2π · (8 in)² + 2π · (8 in) · (12 in)

A = 320π in²

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The greatest common factor of -x^2 + 6x - 19 is 1.

To find the greatest common factor (GCF) of the polynomial [tex]-x^2 + 6x - 19,[/tex] we need to factorize the polynomial and identify the common factors among its terms.

First, let's examine the polynomial and see if we can factor it further:

[tex]-x^2 + 6x - 19[/tex]

The polynomial does not appear to have any common factors among its terms.

It cannot be factored using simple integer factors.

In this case, we can say that the GCF of the given polynomial is 1.

It is important to note that the GCF represents the highest degree of common factors that can be factored out from all terms of a polynomial. In this particular case, the polynomial does not possess any common factors that can be factored out.

It's worth mentioning that this polynomial is already in its simplest form and cannot be factored further.

However, if the polynomial had common factors that could be factored out, such as common binomial factors or other mathematical patterns, the GCF would differ from 1.

But in this case, the polynomial does not exhibit any common factors beyond 1.

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The volume of the solid generated by rotating the region enclosed by the curves y=0, x=2, x=5, and [tex]y=1/\sqrt(1+x)[/tex] about the x-axis is πln(2),

(option A).

To find the volume of the solid generated by rotating the region enclosed by the curves y=0, x=2, x=5, and[tex]y=1/\sqrt(1+x)[/tex] about the x-axis, we need to use the formula for the volume of a solid of revolution.

The formula is ∫[a,b] [tex]π(f(x))^2dx[/tex], where a and b are the limits of integration and f(x) is the function that describes the distance between the axis of revolution and the curve being rotated.

In this case, we need to use the function f(x) = 1/sqrt(1+x), since this function describes the distance between the x-axis and the curve being rotated. The limits of integration are from x=2 to x=5, since these are the values of x that define the region being rotated.

Using the formula, we can set up the integral as follows:

[tex]∫[2,5] π(1/\sqrt(1+x))^2dx[/tex]

= π∫[2,5] 1/(1+x) dx

= π[ln(1+x)]|[2,5]

= π[ln(6) - ln(3)]

= πln(2) (option-A)

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Answer:

(a) x-intercept = -2

(b) y-intercept = 4

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 77th term of the given Sequence is 244.

The given sequence is an arithmetic progression with a common difference of 3. To find the 77th term of the sequence, we can use the formula for the nth term of an arithmetic progression.

The formula for the nth term of an arithmetic progression is:

_ = _1 + ( − 1),

where _ is the nth term, _1 is the first term, is the position of the term in the sequence, and is the common difference.

In this case, _1 = 16 and = 3. Plugging in these values into the formula, we get:

_77 = 16 + (77 − 1) × 3.

_77 = 16 + 76 × 3.

_77 = 16 + 228.

_77 = 244.

Therefore, the 77th term of the given sequence is 244.

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Answer:

The surface area of the hexagonal pyramid is 313.13 square inches.

Step-by-step explanation:

Hexagonal pyramid,

Surface Area = Area of hexagon base + areas of six triangles

b = 6

h = 12.2 (height of the triangle)

A = [tex]\frac{3\sqrt{3} }{2}b + 6(\frac{1}{2} bh)[/tex]

A = [tex]\frac{3\sqrt{3} }{2}b +3bh[/tex]

A = [tex]\frac{3\sqrt{3} }{2}6 +3*6*12.2\[/tex]

A = 93.53 + 219.6

A = 313.13 square inches

Answer: The length of the rectangle is given as "2 feet more than 3 times its width."

Let's break it down step by step:

The formula for the area of a rectangle is length multiplied by width. So, in terms of n, the area of the rectangle would be:

Area = Length × Width

= (3n + 2) × n

= 3n^2 + 2n

Therefore, the area of the rectangle, in terms of n, is 3n^2 + 2n.

We cannot conclude that whether a subject lies or does not lie is independent of the polygraph test indication.

The test statistic needed to test the claim that whether a subject lies or does not lie is independent of the polygraph test indication is the chi-square statistic.

In this case, the grand total is 90. The row totals are 57 and 33, and the column totals are 42 and 48. The expected frequencies are as follows:

The chi-square statistic is calculated as 0.177. The p-value for the chi-square statistic is calculated using a chi-square table. The degrees of freedom for the chi-square table are the number of rows minus 1, multiplied by the number of columns minus 1.

Since the p-value is greater than 0.05, we fail to reject the null hypothesis. We cannot conclude that whether a subject lies or does not lie is independent of the polygraph test indication.

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The length of the cement post's statue is 6.25 feet.

We shall use Mathematical operators to compute the length of the cement post's statue.

(Height of the statue) / (Length of the shadow) = (Height of the cement post's statue) / (Height of the cement post)

Given:

Height of the statue = 25 feet

Length of the shadow = 16 feet

Height of the cement post = 4 feet

Let x =  length of the cement post's statue

25 feet / 16 feet = x / 4 feet

To find x, we cross-multiply:

25 * 4 = 16 * x

100 = 16 * x

To isolate x, we divide both sides of the equation by 16 feet:

100 / 16 = x

x ≈ 6.25 feet

Therefore, the length of the cement post's statue is 6.25 feet.

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The measure of angle C is,

⇒ x = 22°

We have to given that,

In a triangle ABC,

Angle A = 65 degree,

Angle B = 93 degree.

Now, Let us assume that,

Measure of angle C = x

Hence, We get;

⇒ ∠A + ∠B + ∠C = 180°

Substitute all the values,

⇒ 65 + 93 + x = 180

⇒ 158 + x = 180

⇒ x = 180 - 158

⇒ x = 22

Therefore, The measure of angle C is,

⇒ x = 22°

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Complete question is,

In Triangle ABC, Angle A = 65 degree, B=93 degree. Find Angle C​?

For each pound of the blend, Cold Beans should use 3 pounds of Guatemalan coffee.

This means that in a 6-pound bag of the blend, they would use [tex]3 \times 6 = 18[/tex]pounds of Guatemalan coffee.

Let's assume that x pounds of Guatemalan coffee are used per pound of the blend.

Given information:

Cost of Guatemalan coffee = $11/lb

Cost of the other coffee = $5/lb

Desired cost of the blend = $8/lb

Total weight of the blend = 6 pounds

To find the ratio of Guatemalan coffee to the total blend, we can set up the equation:

[tex](x \times 11 + (6 - x) \times 5) / 6 = 8[/tex]

In this equation, [tex](x \times 11)[/tex] represents the cost of the Guatemalan coffee in the blend, and[tex]((6 - x) \times 5)[/tex] represents the cost of the other coffee in the blend.

The numerator is the total cost of the blend, and we divide it by 6 (the total weight of the blend) to find the cost per pound.

Now, let's solve the equation for x:

(11x + 30 - 5x) / 6 = 8

6x + 30 = 48

6x = 48 - 30

6x = 18

x = 18/6

x = 3

Therefore, for each pound of the blend, Cold Beans should use 3 pounds of Guatemalan coffee.

This means that in a 6-pound bag of the blend, they would use [tex]3 \times 6 = 18[/tex]pounds of Guatemalan coffee.

To summarize, to achieve a cost of $8 per pound for a 6-pound bag of blend, Cold Beans should use 3 pounds of Guatemalan coffee per pound of the blend.

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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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Answer:

Volume = 105 cm³

Step-by-step explanation:

Volume of a cuboid = l × b × h (l = length. b = breadth. h = height)

Volume = 7 × 5 × 3

Volume = 35 × 3

Volume = 105 cm³

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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Consecutive Interior Angles.

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

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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