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The Distance from the point P(2, 1, 4) to the plane is approximately -0.9487.

The equation of a plane can be found using the normal vector and a point on the plane.

First, we can find the normal vector of the plane by finding the cross product of two vectors connecting the points on the plane.

The vector from point Q to R is (0, 2, 0) - (1, 0, 0) = (-1, 2, 0)

The vector from point Q to S is (0, 0, 3) - (1, 0, 0) = (-1, 0, 3)

The normal vector is the cross product of these two vectors:

n = cross(-1, 2, 0), (-1, 0, 3)) = (3, 3, -2)

Next, we can find the equation of the plane using the normal vector and a point on the plane, say Q(1, 0, 0):

ax + by + cz = d

3x + 3y - 2z = d

3x + 3y - 2z = 3(1) + 3(0) - 2(0) = 3

Finally, we can find the distance from the point P(2, 1, 4) to the plane by finding the perpendicular distance between the point and the plane. This can be done using the formula:

[tex]d = (Ax + By + Cz + D)/\sqrt{ {A^2 + B^2 + C^2}[/tex]

[tex]d = (3x + 3y - 2z + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]

[tex]d = (3(2) + 3(1) - 2(4) + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]

[tex]d = (-3)/\sqrt{(18)[/tex]

d = -3/3.162 = -0.9487

So the distance from the point P(2, 1, 4) to the plane is approximately -0.9487.

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Invertible matrices P = [1 -2; 0 1] and Q = [1 0; 2 1] such that

A = PQ =  [1 2; 2 -3]  is non-invertible.

To find invertible matrices such that a given matrix is non-invertible, we can use the fact that if A is non-invertible, then the system of linear equations Ax = 0 has a non-trivial solution. This means that there exists a non-zero vector x such that Ax = 0.

Let's start with a non-invertible matrix A, for example:

A = [1 2; 2 4]

The determinant of A is 0, which means that A is non-invertible.

To find a non-zero vector x such that Ax = 0,

We can solve the system of linear equations:

x + 2y = 0

2x + 4y = 0

This system is equivalent to the single equation:

x + 2y = 0

If we choose y = 1, then x = -2, and we get the non-zero vector:

x = [-2; 1]

Now we can use x to construct invertible matrices P and Q such that

PQ = A, as follows:

P = [1 -2; 0 1]

Q = [1 0; 2 1]

The inverse of P is:

P^-1 = [1 2; 0 1]

And the inverse of Q is:

Q^-1 = [1 0; -2 1]

We can verify that P and Q are invertible and that PQ = A:

PQ = [1 -2; 0 1][1 0; 2 1]

PQ = [1 -2; 2 -4 + 1]

PQ = [1 2; 2 -3] = A

Therefore, we have found invertible matrices P and Q such that A = PQ is non-invertible.

Note:- that neither P nor Q is a diagonal matrix, and they are not scalar multiples of each other.

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The complete question may be:

Find non-invertible matrices A, B such that A+B is invertible. Choose

A, B, so that (1) neither is a diagonal matrix and (2) A, B are not scalar multiples of each other.

The amount grace invested $3420 at 7% and the rest of her savings, $5580, at 9%.

Simple interest is a method of calculating the interest charge. Simple interest can be calculated as the product of principal amount, rate and time period.

Simple Interest = (Principal × Rate × Time) / 100

We are given that;

Amount grace invested= 9000 at 9%

Rate for rest of it= 9%

Annual income=741.6

Now,

Let's call the amount Grace invested at 7% "x".

Then the amount she invested at 9% would be "9000 - x", since she invested the rest at the higher rate.

We know that her annual income from the investments was $741.60.

The amount of money she made from the 7% investment would be 0.07x (7% expressed as a decimal multiplied by the amount invested), and the amount of money she made from the 9% investment would be 0.09(9000 - x) (9% expressed as a decimal multiplied by the amount invested).

So we can set up the equation:

0.07x + 0.09(9000 - x) = 741.60

Simplifying and solving for x:

0.07x + 810 - 0.09x = 741.60

-0.02x = -68.4

x = 3420

Therefore, by the given interest rate answer will be $5580, at 9%.

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The three equations that can be used to solve for y, the length of the room, are:

1. y(y + 5) = 750

2. y^2 – 5y = 750

3. (y + 25)(y – 30) = 0

Explanation:

Let's assume that the length of the room is y and the width of the room is y - 5.

We know that the area of the room is the product of its length and width, so we can write an equation:

y(y - 5) = 750

Simplifying this equation, we get:

y^2 - 5y - 750 = 0

Now we can solve this quadratic equation using the quadratic formula or factoring method. By factoring, we can get equation 3. By using the quadratic formula, we can get equation 2. Equation 1 is just another form of equation 2. Therefore, options 1, 2, and 3 can be used to solve for y. Option 4 is not a valid equation as it doesn't represent the area of the room.

The integral for the volume of the solid is ∫2π(eᵇ)(dy)

In calculus, finding the volume of a solid is an important concept. One method to do this is by using integration.

To find the volume of the solid obtained by rotating the region bounded by y=ln(x), y=0, and x=5 about the y-axis, we can use the method of cylindrical shells. This involves dividing the region into infinitely thin vertical strips and rotating each strip around the y-axis to create cylindrical shells.

To set up the integral, we can use the formula for the volume of a cylindrical shell, which is 2πrhΔx, where r is the distance from the y-axis to the edge of the strip, h is the height of the strip, and Δx is the thickness of the strip.

Since we are rotating around the y-axis, we need to express the curves in terms of y instead of x. Using the inverse of the natural logarithm function, we can rewrite

=> y=ln(x) as x=eᵇ.

Thus, the region is bounded by x=5, y=0, and

=> x=eᵇ.

Next, we need to find the distance from the y-axis to the edge of the strip, which is simply the x-value at a given y. This is given by r=e^y.

The height of the strip is simply Δy, the thickness of the strip. To find this, we can take the difference between two consecutive y-values. Thus, Δy=dy.

Putting it all together, the integral for the volume of the solid is:

=> ∫2π(eᵇ)(dy)

This integral represents the sum of the volumes of all the cylindrical shells that make up the solid. However, we still need to evaluate the integral to get the actual volume of the solid.

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The system of equations to represent the situation are f + l = 20.5 and 6.25f + 9.5l = 149.25, solution is 14 pounds and 6.5 pounds

A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.

We are given that;

Cost of fish=$6.25

Cost of lobster=$9.50

Cost of 20.5 pounds =$149.25

Now, let's use the variables f and l to represent the pounds of fish and lobster sold, respectively. We can set up two equations based on the information given:

f + l = 20.5 (the total weight sold is 20.5 pounds)

6.25f + 9.5l = 149.25 (the total revenue made is $149.25)

Now we can solve the system of equations using any method we prefer. Here's one way to do it using substitution:

f + l = 20.5 (equation 1)

f = 20.5 - l (solve equation 1 for f)

6.25f + 9.5l = 149.25 (equation 2)

6.25(20.5 - l) + 9.5l = 149.25 (substitute f = 20.5 - l into equation 2)

128.13 - 6.25l + 9.5l = 149.25 (distribute the 6.25)

3.25l = 21.12 (combine like terms)

l = 6.5 (divide both sides by 3.25)

Now we can use this value of l to find f:

f + 6.5 = 20.5 (from equation 1)

f = 14 (subtract 6.5 from both sides)

Therefore, by the equations the fish market sold 14 pounds of fish and 6.5 pounds of lobster.

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Out of a population of 100,000, the number of people who read at least two newspapers is = 33,000.

Let's approach this problem using the inclusion-exclusion principle.

First, we can add up the proportions of people who read each paper to get:

P(I) + P(II) + P(III) = 10% + 30% + 5% = 45%

However, this includes the people who read two or more papers multiple times, so we need to subtract those out. We can calculate these as follows:

P(I&II) + P(I&III) + P(II&III) = 8% + 2% + 4% = 14%

2P(I&II&III) = 2%

Using the inclusion-exclusion principle, we can now find the proportion of people who read at least two papers:

P(at least 2 papers) = P(I) + P(II) + P(III) - (P(I&II) + P(I&III) + P(II&III)) + 2P(I&II&III)

Plugging in the values, we get:

P(at least 2 papers) = 45% - 14% + 2% = 33%

So, the number of people who read at least two newspapers is:

0.33 * 100,000 = 33,000

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Complete question is:

A certain town of population size 100,000 has three newspapers: I , II and III the proportions of townspeople that read these papers are:  

I= 10 percent

II= 30% percent

II=5 percent

I&II=8 percent

I&III=2 percent

II&III=4 percent

I&II&III=1 percent

How many people read at least two newspapers?

The limit of the difference quotient must not exist at any location z0 in the complex plane in order to demonstrate that f(z) = z is nowhere differentiable.

For f(z), the difference ratio is as follows:

[f(z Plus h) - f(z)] / h = [(z + h) - z] / h = h / h = 1

As h gets closer to 0, we take the maximum and obtain:

lim h0 [z + h - z] / = lim h 0 h / h = 1

This limit is constant at 1 and is unaffected by the number of z. The limit of the difference quotient must not exist at any location z0 in the complex plane in order to demonstrate that f(z) = z is nowhere differentiable. As a result, f(z) = z is never differentiable and the limit of the difference quotient is not present at any position z0 in the complex plane.

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Answer:

Step-by-step explanation:

The zeros of a polynomial function are the values of x for which the function equals zero. For the given polynomial function p(x) = (2x - 1)(5x + 7)(8x + 9), the zeros can be found by setting each factor equal to zero and solving for x:

2x - 1 = 0, x = 1/2

5x + 7 = 0, x = -7/5

8x + 9 = 0, x = -9/8

Therefore, the zeros of the function are x = 1/2, x = -7/5, and x = -9/8.

In standard form, the polynomial function p(x) can be written as:

p(x) = 80x^3 + 122x^2 - 127x - 63

The relationship between the zeros of p(x) and its coefficients can be seen in Vieta's formulas. Vieta's formulas state that for a polynomial function of degree n with roots r1, r2, ..., rn, the coefficients of the polynomial can be expressed as:

a0 = (-1)^n * p0

a1 = (-1)^(n-1) * p1 / p0

a2 = (-1)^(n-2) * p2 / p0

...

an-1 = (-1) * pn-1 / p0

an = pn / p0

where p0 is the coefficient of the highest degree term (the leading coefficient), and the pi are the elementary symmetric polynomials, which are given by:

p1 = r1 + r2 + ... + rn

p2 = r1r2 + r1r3 + ... + rn-1rn

...

pn-1 = r1r2...rn-1 + r1r2...rn-2 + ... + rn-2rn-1

pn = r1r2...rn

Using Vieta's formulas, we can see that for the polynomial function p(x) given above, the coefficients are related to the zeros as follows:

a0 = -63

a1 = -127

a2 = 122

a3 = 80

And we can also see that:

a0 = (-1)^3 * p0 = -63

a1 = (-1)^(3-1) * p1 / p0 = -127

a2 = (-1)^(3-2) * p2 / p0 = 122

a3 = (-1)^(3-3) * p3 / p0 = 80

Therefore, the coefficients of the polynomial are related to the zeros through Vieta's formulas, which express the coefficients as functions of the zeros, and vice versa.

Box plot is a chart that shows data from a five-number summary including one of the measures of central tendency.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

A box plot, also known as a box and whisker plot, is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.

The box plot consists of a rectangular box with whiskers that extend from the edges of the box.

The box represents the middle 50% of the data, and the vertical line inside the box represents the median (50th percentile) of the data.

The whiskers extend to the smallest and largest observations that are within 1.5 times the interquartile range (IQR) of the lower and upper quartiles, respectively.

Any observations that fall outside the whiskers are considered outliers and are plotted as individual points.

Box plots are useful for comparing the distribution of data between different groups of data.

Hence, Boxplot is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.

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The statements about the given situation that are true about domain and range  are;

B: The maximum value in the range is $320.

D: The minimum value in the domain is 0.

The domain is defined by b, the number of bracelets sold.

The minimum value in the domain is 0, which represents no bracelets sold.

The maximum value in the domain is 260, which represents the largest number of bracelets the group can make, and the largest number they could sell.

The range is defined by f(b), the amount of profit on the bracelets.

To find the maximum value in the range, we find f(260), the profit on selling the maximum in the domain.

Substitute 260 for b in f(b) = 2b – 200 to get:

f(260) = 2(260) – 200

f(260) = 320

The maximum value in the range is $320.

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Complete question is;

The high school jazz band is selling homemade leather bracelets at a local craft fair to raise money for a trip. The group has a $200 budget to spend on supplies, which is enough to make 260 bracelets. The group is charging $2 per bracelet at the craft fair.

Which statements about this situation are true?

Select all the correct answers.

The maximum value in the range is $200.

The maximum value in the range is $320.

The maximum value in the domain is 200.

The minimum value in the domain is 0.

False. The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane.

The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane. This equation can be written as [tex]x = x2u + x3v[/tex], which is the same as [tex]ux2 + vx3 - x = 0[/tex]. This equation is in the form of [tex]ax + by + cz = d[/tex], with a = u, b = v, c = -1, and d = 0. Since the coefficients of [tex]x2, x3[/tex], and x are all non-zero, we can conclude that this equation describes a line, not a plane. To confirm, we can calculate the direction vector for the line. The direction vector is (u, v, -1), which is a single vector and not two or more vectors that would be necessary to describe a plane.

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Answer:

y= 2x+12

Step-by-step explanation:

y=mx+c

y= 2x+12

(0,y) (-4,4)

2=y-4/4

8=y-4

12=y

Answer:m=2/5

Step-by-step explanation:

m[tex]\neq[/tex]0

5/6=1/3m

15m=6

m=2/5, m[tex]\neq[/tex]0

Answer:

A)  Vertex = (-3, 4), y-intercept = (0, -5), x-intercepts = (-5, 0) and (-1, 0), axis of symmetry is x = -3

Step-by-step explanation:

The vertex of a quadratic function is the turning point. As this parabola opens downwards, the vertex is the maximum point of the graph. From inspection of the graph, the maximum point is at (-3, 4). Therefore:

The y-intercept is the point at which the curve crosses the y-axis. From inspection of the graph, the curve crosses the y-axis at y = -5. Therefore:

The x-intercepts are the points at which the curve crosses the x-axis. From inspection of the graph, the curve crosses the x-axis at x = -5 and x = -1. Therefore:

The axis of symmetry is the vertical line that passes through the vertex of the parabola so that the left and right sides of the parabola are symmetrical. So the axis of symmetry is the x-value of the vertex. Therefore:

We can use the Law of Cosines to solve for angle C:

cos C = (a^2 + b^2 - c^2) / (2ab)

where b is the length of side BC.

We know that B = 40 degrees, so we can find the measure of angle A:

A = 180 - B - C

A = 180 - 40 - C

A = 140 - C

We also know that the sum of the angles in a triangle is 180 degrees, so:

A + B + C = 180

140 - C + 40 + C = 180

180 = 180

This checks out, so we can continue with solving for angle C:

cos C = (a^2 + b^2 - c^2) / (2ab)

cos C = (7^2 + b^2 - 10^2) / (2*7b)

cos C = (49 + b^2 - 100) / (14b)

cos C = (b^2 - 51) / (14b)

We can use the Law of Sines to solve for b:

sin B / b = sin A / a

sin 40 / b = sin (140 - C) / 7

b = sin 40 * 7 / sin (140 - C)

Now we can substitute this value of b into the equation for cos C:

cos C = (b^2 - 51) / (14b)

cos C = [(sin 40 * 7 / sin (140 - C))^2 - 51] / (14 * sin 40 * 7 / sin (140 - C))

Simplifying this equation and solving for cos C, we get:

cos C = 11/14

Therefore, angle C is:

C = cos^-1(11/14)

C ≈ 38.8 degrees

Answer:

24.5in

Step-by-step explanation:

7 days in a week. 7 weeks. 7 x 7=49

49 x 0.5 + 24.5in

The required scaled copy of polygon B using a scale factor of 0.75 as shown.

Scale image is defined as a ratio that represents the relationship between the shape and size of a figure and the corresponding dimensions of the actual figure or object.

The polygon B is given as shown, which dimensions are below:

Length of polygon = 8 units

Height of polygon = 10 units

Here, the scale factor = 0.75

So, the dimensions of the scaled copy of polygon B are below:

Length of polygon B' = 8 × 0.75 = 6 units

Height of polygon B' = 10 × 0.75 = 7.5 units

Thus, the scaled copy of polygon B using a scale factor of 0.75 as shown.

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Leah's argument is that the rounded up figures for Tuesday sales are greater than the sales for Monday. She supports it by summing up the sales figures.

Leah's reasoning is wrong because she rounded up $ 233 to $ 300 instead of to $ 200.

Leah's reasoning therefore does not make sense.

Leah attempts to round the Tuesday sales figures to the nearest 100. In doing so, she rounded $ 233 to $ 300 instead of $ 200 which was the closest.

If she had done so, the result would be :

= 200 + 400

= $ 600

This would then show that Monday's figures were higher than Tuesday.

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Answer:

$381

Step-by-step explanation:

9% = 1 year

27% = 3 years

$300 = 100%

After 3 years, we have

100% + 27% = 127%

127% = 1.27

300 tines 1.27 = $381

So, after 3 years has $381

With the sum of digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. The number is 18. So, the correct option is A.

Let the two-digit number be represented by 10x + y, where x is the tens digit and y is the ones digit. We are told that x + y = 9 and that if the digits are reversed, the number is increased by 63.

If we reverse the digits of the original number, we get 10y + x. The difference between this number and the original number is 63, so we can set up the equation:

10y + x - (10x + y) = 63

Simplifying this equation, we get:

9y - 9x = 63

Dividing both sides by 9, we get:

y - x = 7

Now we have two equations: x + y = 9 and y - x = 7. We can solve this system of equations by adding the two equations:

2y = 16

y = 8

Substituting y = 8 into x + y = 9, we get:

x + 8 = 9

x = 1

Therefore, the original number is 10x + y = 10(1) + 8 = 18. So the answer is (a) 18.

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The polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).

A polynomial is a mathematical expression which have a sum of powers in one or more variables with coefficients. The highest power of the variable in a polynomial is called its degree.

We shall divide the polynomial x³ - 4x³ + 5x² - 2 by x - 1 as follows;

x³ divided by x equals x²

x - 1 multiplied by x² equals x³ - x²

subtract x³ - x² from x³ - 4x³ + 5x² - 2 will result to -3x² + 5x - 2

-3x² divided by x equals -3x

x - 1 multiplied by -3x equals -3x² + 3x

subtract -3x² + 3x from x³ - 4x³ + 5x² - 2 will result to 2x - 2

2 divided by x equals 2

x - 1 multiplied by 2 equals 2x - 2

subtract 2x - 2 from 2x - 2 will result to a remainder 0

Therefore, the polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).

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Answer:

The United States form of government is a...

O Republic

can I please get the five points:)

Answer:

Republic

Step-by-step explanation:

Option B

I hope this helps :) if not let me know

The distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.

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

Distance, D = 150 km

Rates = 35 km/h and 30 km/h

Let t be the time elapsed from noon to 4:00 PM

So, we have

t = 4

The distance between the ships to their distances is represented as

d^2 = (D + rate 1 * t)^2 + (rate 2 * t)^2

So, we have

d^2 = (150 + 35t)^2 + (30t)^2

d^2 = (150)^2 + 10500t + 1225t^2 + 900t^2

Differentiate with respect to time (t)

2D d' = 10500 + 2450t + 1800t

So, we have

D d' = 5250 + 1225t + 900t

Substitute 4 for t

D d' = 13750

So, we have

d' = 13750/D

This gives

d' = 13750/150

d' = 91.6

So the distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.

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The value of x is 9.

The value of y is 5.

An equation contains one or more terms with variables connected by an equal sign.

Example:

2x + 4y = 9 is an equation.

2x = 8 is an equation.

We have,

4x - 2y = 26

This can be written as,

4x - 26 = 2y ______(1)

5x – 2y = 35

This can be written as,

5x - 35 = 2y _______(2)

From (1) and (2),

4x - 26 = 5x - 35

35 - 26 = 5x - 4x

9 = x

x = 9

And,

Substituting x = 9 in (1),

4x - 26 = 2y

4 x 9 - 26 = 2y

36 - 26 = 2y

2y = 10

y = 5

Thus,

The solution is (9, 5).

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Answer:8

Step-by-step explanation:

15/15=1

112/15=7.466...

you can't buy 7.466... cups, so you round up to 8.

hope I'm right ;)

The domain of the function is [0, 3) and the range of the function is [-4, 5).

A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.

Here the input values in set A is called the domain and the output values in set B is called the range.

Given is a graph of a function.

The input values or domain includes the value of x which forms the curve in the graph.

The output values or range consists of the values of y formed as a result of the input of x.

Look at the x values corresponding to the end points of curve.

The curve extends from x = 0 to x = 3.

But at x = 3, it is an open circle. So x = 3 is not included.

So the input values are from x = 0 to x = 3, where 3 is not included.

Domain in interval notation = [0, 3).

Similarly the values of y extends from y = -4 to y = 5, where 5 is not included, since there is an open circle there.

Range in interval notation = [-4, 5).

Hence the domain and range are [0, 3) and [-4, 5) respectively.

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The lateral surface area of a prism is the sum of the areas of all its rectangular faces.

Surface area is a two-dimensional measure that refers to the total area of a surface, such as the area of a two-dimensional shape, a three-dimensional solid, or a combination of both. It is the sum of the areas of all the faces of a solid object. It is also referred to as the area of the boundary of a three-dimensional object. It can be used to calculate the volume of an object, and is also used in other calculations like area of the base of a triangle, area of a circle, and more.

It is calculated by multiplying the perimeter of the base by the height of the prism. For example, if the base of a prism is a square with side length s and the height is h, then the lateral surface area of the prism can be calculated as 4sh. If the base of the prism is a rectangle with length l and width w, then the lateral surface area of the prism can be calculated as 2lw + 2wh.

The lateral surface area of a prism can also be calculated by adding the areas of the triangular faces. If the base of the prism is a triangle with side lengths a, b, and c, then the lateral surface area of the prism can be calculated as a + b + c.

It is important to note that the lateral surface area of a prism is different from its surface area. The surface area of a prism is the sum of the areas of its lateral faces and its two bases.

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Complete question:

How to find the lateral surface area of the prism?

Answer:

  y = -5

Step-by-step explanation:

You want to solve for y in 2y -3(2y -3) +2 = 31.

Parentheses can be eliminated using the distributive property.

  2y -6y +9 +2 = 31

Like terms can be combined.

  -4y +11 = 31

We can separate the constant and variable terms by subtracting 11 from both sides.

  -4y = 20

The value of y is now found by dividing by -4.

  y = 20/(-4) = -5

  y = -5