What’s the volume of this triangular prism ?

No squares should be cut. Maximum volume is zero. Dimensions remain 15cm by 40cm by 0cm.

To find the size of the square that should be cut out from each corner to maximize the volume of the bin, we need to follow these steps:

1. Understand the problem : We have a rectangular piece of plastic with dimensions 15 cm by 40 cm. By removing squares from each corner and folding up the flaps, we can form an open-top bin. We want to find the size of the squares to be cut out to maximize the volume of the bin.

2. Define variables : Let's denote the side length of the square to be cut out from each corner as [tex]\( x \)[/tex] cm.

3. Express the volume of the bin : After cutting out squares and folding up the flaps, the dimensions of the resulting bin will be [tex]\( (15 - 2x) \)[/tex] cm by [tex]\( (40 - 2x) \)[/tex] cm by [tex]\( x \)[/tex] cm. So, the volume of the bin can be expressed as:  [tex]\[ V = x(15 - 2x)(40 - 2x) \][/tex]

4. Maximize the volume: To find the maximum volume, we need to find the critical points of the function [tex]\( V \)[/tex]. We'll take the derivative of [tex]\( V \)[/tex] with respect to [tex]\( x \)[/tex], set it equal to zero, and solve for [tex]\( x \)[/tex].

  [tex]\[ \frac{dV}{dx} = (15 - 2x)(40 - 2x) + x(-4)(40 - 2x) + x(15 - 2x)(-4) \][/tex]

  [tex]\[ 0 = (15 - 2x)(40 - 2x) - 8x(40 - 2x) - 4x(15 - 2x) \][/tex]

5. Solve for [tex]\( x \)[/tex] : Solve the equation for [tex]\( x \)[/tex] to find the critical points.

 [tex]\[ 0 = 600 - 70x + 4x^2 \][/tex]

 [tex]\[ 0 = 4x^2 - 70x + 600 \][/tex]

  Solve for [tex]\( x \)[/tex] using the quadratic formula:

  [tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

  where [tex]\( a = 4 \)[/tex], [tex]\( b = -70 \)[/tex], and [tex]\( c = 600 \)[/tex].

  [tex]\[ x = \frac{-(-70) \pm \sqrt{(-70)^2 - 4(4)(600)}}{2(4)} \][/tex]

  [tex]\[ x = \frac{70 \pm \sqrt{4900 - 9600}}{8} \][/tex]

  [tex]\[ x = \frac{70 \pm \sqrt{-4700}}{8} \][/tex]

  Since the discriminant is negative, there are no real solutions for [tex]\( x \)[/tex] in this case. Therefore, we need to check the endpoints of our interval.

6. Check endpoints : Since [tex]\( x \)[/tex] must be non-negative and [tex]\( (15 - 2x) \)[/tex] and [tex]\( (40 - 2x) \)[/tex] must also be positive, the possible range for [tex]\( x \)[/tex] is [tex]\( 0 \leq x \leq 7.5 \)[/tex] cm.

  When [tex]\( x = 0 \), \( V = 0 \)[/tex] (no bin is formed).

  When [tex]\( x = 7.5 \), \( V = 7.5(15 - 2(7.5))(40 - 2(7.5)) = 7.5(0)(25) = 0 \)[/tex] (no bin is formed).

7. Conclusion : The maximum volume of the bin is achieved when no squares are cut out from the corners. Therefore, to maximize the volume, no squares should be cut out from the corners of the plastic, resulting in a rectangular prism with dimensions [tex]\( 15 \times 40 \times 0 \)[/tex] cm³, which does not form a bin.

So, the maximum volume of the bin is [tex]1000cm^3[/tex]

To maximize the volume of the bin, we need to determine the dimensions of the squares that should be cut out from each corner to form the bin. Let's denote the side length of the square to be cut out as

x centimeters.

When the squares are cut out and the flaps are folded up, the dimensions of the resulting bin can be expressed as follows:

Length of the bin: 40−2x centimeters

Width of the bin: 15−2x centimeters

Height of the bin (equal to the side length of the square cut out): x centimeters

The volume V of the bin is given by the product of its length, width, and height:

V=(40−2x)(15−2x)x

To find the maximum volume, we need to find the value of

x that maximizes this function. We can do this by taking the derivative of

V with respect to x, setting it equal to zero, and solving for x.

Let's find the derivative of V with respect to x:

[tex]$\frac{d V}{d x}=15(40-2 x)-2(15-2 x)(40-2 x)$[/tex]

Now, let's set the derivative equal to zero and solve for x:

[tex]$15(40-2 x)-2(15-2 x)(40-2 x)=0$[/tex]

After solving for x, we can substitute the value of

x back into the expression for the volume

V to find the maximum volume of the bin. Let's do these calculations.

First, let's simplify the derivative[tex]\frac{dv}{dx}[/tex]

[tex]$\begin{aligned} & \frac{d V}{d x}=15(40-2 x)-2(15-2 x)(40-2 x) \\ & =600-30 x-\left(30 x-4 x^2\right) \\ & =600-30 x-30 x+4 x^2 \\ & =600-60 x+4 x^2\end{aligned}$[/tex]

Now, we set the derivative equal to zero and solve for x:

[tex]$600-60 x+4 x^2=0$[/tex]

Dividing both sides by 4:

[tex]$150-15 x+x^2=0$[/tex]

Rearranging and factoring:

[tex]$$\begin{aligned}& x^2-15 x+150=0 \\& (x-10)(x-15)=0\end{aligned}$$So, $x=10$ or $x=15$.[/tex]

Since x=15 would result in a negative side length for the bin, we discard it.

Thus, the optimal size square to be cut out from each corner is 10 centimeters.We can then find the maximum volume

V by substituting x=10 into the volume formula:

[tex]$V=(40-2 \times 10)(15-2 \times 10) \times 10=(20)(-5) \times 10=1000 \mathrm{~cm}^3$[/tex]

So, the maximum volume of the bin is [tex]1000cm^3[/tex]

Answer:

Dimensions of page should be width of 6 inches and height of 9 inches

Step-by-step explanation:

Let x be the width of the printed part in inches

Let y be height of the printed part in inches.

Thus, Area of printed part; A = xy

And area of printed part is given as 24.

Thus, xy = 24

Making y the subject, we have;

y = 24/x

Now, the question says the top and bottom margins are 1.5 inches.

Thus, width of page = x + 1 + 1 = x + 2

And also the margins on each side are both 1m in length, thus the height of page will be:

y + 1.5 + 1.5 = y + 3

So area of page will now be;

A = (x + 2)•(y+3)

From earlier, we got y = 24/x

Thus,plugging this into area of page, we have;

A = (x + 2)•((24/x)+3)

A = 24 + 3x + 48/x + 6

A = 30 + 3x + 48/x

For us to find the minimum dimensions, we have to find the derivative of A and equate to zero

Thus,

dA/dx = 3 - 48/x²

Thus, dA/dx = 0 will be

3 - 48/x² = 0

Multiply through by x²:

3x² - 48 = 0

Thus,

3x² = 48

x² = 48/3

x = √16

x = 4 inches

Plugging this into y = 24/x,we have;

y = 24/4 = 6 inches

We want dimensions of page at x = 4 and y = 6.

From earlier, width of page = x + 2.

Thus,width = 4 + 2 = 6 inches

Height = y + 3 = 6 + 3 = 9 inches

So dimensions of page should be width of 6 inches and height of 9 inches

To minimize paper usage, we need to find the dimensions of the rectangular page. By setting up an equation and solving it, we can find the width and length of the page that result in the least amount of paper used.

To minimize the amount of paper used, we need to find the dimensions of the page that will result in the smallest possible area. Let's assume the length of the page is 'L' inches and the width is 'W' inches. We can set up the equation:

Next, we can use the quadratic formula to solve for L:

To find the minimum value of L, we can consider the factors contributing to the uncertainty:

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Answer:

The lengths of time that his mom reads are typically longer and have more variability than the lengths of time that his dad reads.

Step-by-step explanation:

The Mom's plot is longer in ranges, therefore, she would have more variability than the dad.

Answer:

B

Step-by-step explanation:

Answer:

The answer is 18 feet squared

Step-by-step explanation:

Answer:

The answer is 18 ft squared

Step-by-step explanation:

i just got 100% i know all the right answers

Answer:

45.216‬

Step-by-step explanation:

2 x 3.14 x7.2

Answer:

Step-by-step explanation:

Volume of water bottle = 143 cubic cm

πr²h = 143

3.14*r²*7.2 = 143

[tex]r^{2}=\frac{143}{3.14*7.2}\\\\r^{2}=6.325\\[/tex]

r = 2.51

r = 2.5 cm

Answer:

The surface area of the sphere is 86525.84 square yards

Step-by-step explanation:

The widest circle of the sphere has the same radius of the sphere

Use the circumference of the circle to find its radius, then use it to find the surface area of the sphere

The formula of the circumference of the circle is C = 2π r

The formula of the surface area of the sphere is SA = 4π r²

∵ The circumference of the widest circle is 521.24 yards

∴ C = 521.24

- Equate the formula of the circumference by it

∴ 2π r = 521.24

∵ π ≈ 3.14

∴ 2(3.14) r = 521.24

∴ 6.28 r = 521.24

- Divide both sides by 6.28

∴ r = 83 yards

Now use the formula of the surface area of the sphere to find it

∵ The radius of the sphere = the radius of the widest circle

∴ The radius of the sphere is 83 yards

∵ SA = 4π r²

- Substitute r by 83 and π by 3.14

∴ SA = 4(3.14)(83)²

∴ SA = 86525.84

The surface area of the sphere is 86525.84 square yards

Answer:

A = π(7x + 3)² cm²

Step-by-step explanation:

A = πr² is the appropriate equation.  If r = 7x + 3 cm, then the area of this particular circle is:

A = π(7x + 3)² cm²

Final answer:

The area of a circle with radius (7x+3)cm is represented by the simplified expression 49x²π + 42xπ + 9π square centimeters.

Explanation:

To find the area of a circle with radius (7x+3)cm, we use the formula for the area of a circle, which is A = πr². So, substituting the given radius into this formula, we get:

A = π(7x+3)²

To simplify the expression, we square the binomial:

A = π(49[tex]x^2[/tex] + 42x + 9)

Therefore, the area of the circle in simplified form is 49[tex]x^2[/tex]π + 42xπ + 9π square centimeters.

Answer:

  a.  csc²θ

Step-by-step explanation:

You can use the identities ...

  1 +tan² = sec²

  cot = cos/sin

  sec = 1/cos

  csc = 1/sin

___

Then the expression becomes ...

  [tex]\cot^2{\theta}(1+\tan^2{\theta})=\cot^2{\theta}\sec^2{\theta}=\dfrac{\cos^2{\theta}}{\sin^2{\theta}\cos^2{\theta}}=\dfrac{1}{\sin^2{\theta}}\\\\=\boxed{\csc^2{\theta}}[/tex]

Step-by-step explanation:

Putting value of y

-8x = 6x - 14

-8x - 6x = - 14

-14x = - 14

x = - 14/-14

x = 1

Answer:

V(cylinder) = πr²h

h=3 cm

r=2 cm

V(cylinder) = πr²h = π*2²*3 =12π cm³

Answer : 12π cm³.

Answer:

A

Step-by-step explanation:

12π cm³

x = dimes

y = quarters

We also know:

Dimes: 0.10

Quarters: 0.25

100 = x + y

So, 0.10x + 0.25y = 100

And, y = 100 - x

Let’s use substitution:

0.10x + 0.25( 100 - x ) = 100

= 0.10x + 25 - 0.25x = 100

= 25 - 100 = 0.25x - 0.10x

= (-75) = 0.15x

x = (-75) divide by 0.15

x = (-500)

Answer: the minimum score required for a letter of recognition is 27.8

Step-by-step explanation:

Suppose ACT Mathematics scores are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = ACT Mathematics scores.

µ = mean score

σ = standard deviation

From the information given,

µ = 21.3

σ = 5.3

The probability of students whose scores are in the top 11 % would be be (1 - 11/100) = (1 - 0.11) = 0.89

Looking at the normal distribution table, the z score corresponding to the probability value is .23

Therefore,

1.23 = (x - 21.3)/5.3

Cross multiplying by 8.6, it becomes

1.23 × 5.3 = x - 21.3

6.519 = x - 21.3

x = 6.519 + 21.3

x = 27.8 to the nearest whole number

Final answer:

The minimum ACT Mathematics score required to be in the top 11% with a mean of 21.3 and a standard deviation of 5.3 is approximately 27.85, which is usually rounded to 28.

Explanation:

Top 11% for ACT Mathematics Score Requirement

If ACT Mathematics scores are normally distributed with a mean of 21.3 and a standard deviation of 5.3, the score corresponding to the top 11% can be found by determining the z-score that corresponds to the 89th percentile (100% - 11% = 89%). To find this z-score, one would typically use a z-score table or a statistical software. Once the z-score for the 89th percentile is found, it can be converted to an ACT score using the formula:

Z = (X - μ) / σ

where Z is the z-score, X is the ACT score, μ (mu) is the mean, and σ (sigma) is the standard deviation. Solving for X:

X = Z×σ + μ

Assuming the z-score for the 89th percentile is approximately 1.23, the minimum required score for a letter of recognition would be:

X = 1.23×5.3 + 21.3

X ≈ 1.23×5.3 + 21.3

X ≈ 6.549 + 21.3

X ≈ 27.849

Therefore, the minimum score required for a letter of recognition would be approximately 27.85, which can be rounded as appropriate for the context (usually to the nearest whole number, giving a score of 28).

Answer:

a

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

taking the square root of both sides of the equation

Answer:

  45

Step-by-step explanation:

The average of the three numbers is the middle one: 138/3 = 46. Then the smallest of the three consecutive numbers is 45.

Final answer:

The smallest of the three consecutive numbers that add up to 138 is 45.

Explanation:

To find the smallest of the three consecutive numbers that add up to one hundred thirty-eight, we first need to represent the numbers algebraically.

Let x be the smallest number, x+1 is the next number, and x+2 is the largest number.

The equation that represents their sum is:

x + (x + 1) + (x + 2) = 138

To solve for x, we combine like terms:

3x + 3 = 138

Then we subtract 3 from both sides:

3x = 135

And divide both sides by 3 to find x:

x = 135 / 3

x = 45

Therefore, the smallest number is 45.

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

9 + 3 = 12

[tex]Water=\frac{9}{12}*48\\\\=9*4[/tex]

Water = 36 liters

Pesticide = 48 - 36  = 12 liters

Answer:

The bottom equation is infinite solutions ( please give branliest)

2x - 6y = 18

x - 3y = 9

Step-by-step explanation:

Using substitution,

x=3y + 9

6x - 6x + 18 = 18

0 = 18 - 18

0 = 0

Answer: Last one:)

Step-by-step explanation:

Just did it

Answer: D

Step-by-step explanation:

The correct answer is d) No because their product is not equal to the identity matrix.

The matrices are given in the question as follows:

[tex]A= \left[\begin{array}{ccc}4&2\\-11&-6\end{array}\right] \\B= \left[\begin{array}{ccc}-4&-2\\11&6\end{array}\right][/tex]

To determine if two matrices are inverses of each other, we need to check if their product is equal to the identity matrix.

Let's calculate the product of matrices A and B:

[tex]A{\times}B = \left[\begin{array}{ccc}4\times-42+2\times11&4\times-2+2\times6\\-11\times-4+(-6\times11)&-11\times-2-6\times6\end{array}\right][/tex]

Simplifying the calculations to get:

[tex]A{\times}B = \left[\begin{array}{ccc}6&4\\-22&-14\end{array}\right][/tex]

The resulting matrix is not equal to the identity matrix:

[tex]A{\times}B = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right][/tex]

Therefore, matrices A and B are not inverses of each other.

The correct answer is d) No because their product is not equal to the identity matrix.

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Answer:

6 times

Step-by-step explanation:

48/8=6

Answer: 10 seconds for one time around the track.

Step-by-step explanation:

48 times=8 minutes

1 time= 8/48

8/48=4/24

4/24=2/12

1/6. 1/6 of a minute equals 10 seconds.

The answer is 10 seconds

Final answer:

Three segments with lengths 4 cm, 6 cm, and 11 cm cannot form a triangle because the sum of the lengths of the two shorter sides is not greater than the length of the longest side, violating the Triangle Inequality Theorem.

Explanation:

To determine whether three segments with lengths 4 cm, 6 cm, and 11 cm can form a triangle, we must consider the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. If we attempt to apply this to the given lengths:

4 cm + 6 cm > 11 cm (False)

4 cm + 11 cm > 6 cm (True)

6 cm + 11 cm > 4 cm (True)

Since the sum of the lengths of the two shorter sides (4 cm + 6 cm) is not greater than the length of the longest side (11 cm), these three segments cannot form a triangle, whether it is an acute triangle, right triangle, or obtuse triangle.

If they were able to form a triangle, to classify the type of triangle, we would use the Converse of the Pythagorean Theorem. This theorem helps us understand if a triangle is acute, right, or obtuse:

In a right triangle, the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.

If the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse.

If the square of the longest side is less than the sum of the squares of the other two sides, the triangle is acute.

However, since a triangle cannot be formed with these segments, we do not need to apply these rules here

Using stratified sampling concepts, it is found that the correct options are:

Stratified sampling divides the population into groups, called stratas, and a few elements of each group are surveyed.

In this problem, the households are divided to represents various​ locations, ethnic​ groups, and income brackets, hence, those are the stratas.

Stratified sampling is used because ensures that all segments of the population are represented.

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Answer:

The answer is 126!!! Hope I helped!!!

Step-by-step explanation:

1. 6 x 3 = 18

2. 18 x 7 = 126

Answer:

126

Step-by-step explanation:

Multiply 7 x 6 x 3 which is equal to 126.

Have a great day!  Your amazing. I do a lot of answers on edg 6th grade.

Feel free to ask questions

Answer:

Step-by-step explanation:

Given

Compare the given with the options to identify equivalents

A)

Not equivalent

B)

Not equivalent

C)

Not equivalent

D)

Equivalent

Final answer:

To find the coordinates of point C that partitions the segment from A(-8,4) to B(10,-2) in the ratio 2:1, we use the formula for internal division of a line segment, resulting in the coordinates (4, 0).

Explanation:

The coordinates of point C that partitions the segment from A(-8,4) to B(10,-2) in the ratio 2:1 can be found using the formula for internal division of a line segment. Since AC:CB is 2:1, we consider the section ratio to be 2/3 and multiply it by the difference in the corresponding x and y coordinates of A and B, then add to the coordinates of point A.

The x-coordinate of point C is calculated as follows:
Cx = Ax + (Bx - Ax) * m/(m+n)
Where Ax is the x-coordinate of point A, Bx is the x-coordinate of point B, and m/n is the ratio (2/1 in this case), so m=2 and n=1.
Cx = (-8) + (10 - (-8)) * 2/(2+1) = -8 + 18 * 2/3 = -8 + 12 = 4

The y-coordinate of point C is calculated similarly:
Cy = Ay + (By - Ay) * m/(m+n)
Cy = 4 + ((-2) - 4) * 2/(2+1) = 4 + (-6) * 2/3 = 4 - 4 = 0

Thus, the coordinates of point C are (4, 0).

Answer:

-6

Step-by-step explanation:

3x * 2 = 6x - 1 = -6

The required result of the given expression is 2x - 5.

An expression is a number, or a variable, or a combination of numbers and variables and operation symbols.

Now the given expression is,

Add 3 to x, double what you have, then subtract 1

So, we can write,

Add 3 to x =  3 + x

Double = 2(3 + x)

Subtract from  = 1 - 2(3 + x)

Simplifying we get,

1 - 6 + 2x

solving we get,

1 - 6 + 2x =  2x - 5

this is the required result.

Thus, the required result of the given expression is 2x - 5.

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Answer: $125

Step-by-step explanation:

1500/12=125

Answer

25 percent

Step-by-step explanation:

Answer:

The correct answer is 100%

Step-by-step explanation:

:>

Given:

The angle of elevation from the point on the ground to the top of the tree is 34° and the point is 25 feet from the base of the tree.

We need to determine the height of the tree.

Height of the tree:

Let the height of the tree be h.

The height of the tree can be determined using the trigonometric ratio.

Thus, we have;

[tex]tan \ \theta=\frac{opp}{adj}[/tex]

Substituting the values, we get;

[tex]tan \ 34^{\circ}=\frac{h}{25}[/tex]

Multiplying both sides by 25, we have;

[tex]tan \ 34^{\circ} \times 25=h[/tex]

 [tex]0.6745 \times 25=h[/tex]

       [tex]16.8625=h[/tex]

Rounding off to the nearest tenth of a foot, we get;

[tex]16.9=h[/tex]

Thus, the height of the tree is 16.9 feet.

Hence, Option B is the correct answer.

Correct option is B. The height of the tree can be found using the tangent of the angle of elevation, which is the ratio of the tree's height to the distance from the base. Applying the formula, the height is approximately 16.9 feet, matching option B.

To calculate the height of the tree given the angle of elevation and the distance from the point on the ground to the base of the tree, we can use trigonometric functions, specifically the tangent function in a right triangle. The tangent of an angle in a right triangle is the ratio of the opposite side (height of the tree) to the adjacent side (distance from the point on the ground to the tree).

Using the formula:

tangent(angle) = opposite / adjacent

Substitute the given values:

tangent(34°) = height of the tree / 25 ft

Now solve for the height:

height = 25 ft * tangent(34°)

Use a calculator to find:

height = 25 ft * 0.6745

height = 16.9 ft

Therefore, the height of the tree is approximately 16.9 feet (to the nearest tenth of a foot), making option B the correct answer.

Explanation: Probability of NOT getting a tail in 3 coin toss is (12)3=18 . Probability of getting at least 1 tail in 3 coin toss is 1−18=78