Algebraic Manipulation: Simplify 15x³ (y-z)² ÷ 10x⁻²(y²-z²)

Thank you!

Answer:

132.5 cm²

Step-by-step explanation:

A cuboid has 3 pairs of identical faces.

So the surface area is 2(L×W) + 2(L×H) + 2(W×H)

the calculation is 34.5 + 75 + 23 = 132.5 cm²

The total surface area of a cuboid with dimensions 7.5cm, 2.3cm, and 5cm is 132.5 cm², calculated using the formula for the surface area of a cuboid being 2lw + 2lh + 2wh.

The subject of this question concerns the calculation of the total surface area of a cuboid. A cuboid has six rectangular faces. To find the total surface area of the cuboid, we need to calculate the area of all six faces. The formula to find the surface area of a cuboid is 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.

In this case, we substitute the given dimensions into the formula to get: 2(7.5)(2.3) + 2(7.5)(5) + 2(2.3)(5) = 34.5 + 75 + 23 = 132.5 cm².

The total surface area of the cuboid is therefore 132.5 cm².

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The answer is 11 and 6.

Hope this helps!

The two numbers which sum up to 17 and have a product of 66 are 6 and 11. This was found by using algebra to set up and solve two simultaneous equations.

The question can be resolved using a little bit of algebra. Let's assign the values x and y to these two numbers. We know two things: x + y = 17 (because their sum is 17) and xy = 66 (because their product is 66).

First, you will make y the subject of the first equation making it y = 17 - x. Replace y in the second equation with 17 - x so we will now have x(17 - x) = 66 or 17x - x^2 = 66. Rearranging the equation gives x^2 - 17x + 66 = 0. This equation can be factored to solve for x giving (x - 11)(x - 6) = 0. Therefore, x could be base on the equation 11 or 6.

If x is 11, y will be 6 (because 17 - 11 = 6) and if x is 6, y is 11 (17 - 6 = 11). Therefore, the two numbers are 6 and 11.

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Answer:

[tex]\dfrac{21}{32}=0.65625[/tex]

Step-by-step explanation:

If the probability of success is 50%, then p=0.5 and q=1-0.5=0.5.

At least three successes in six trials of a binomial experiment means that favorable are 3 successes, 4 successes, 5 successes and 6 successes.

1. 3 successes:

[tex]Pr_1=C^3_6p^3q^{6-3}=\dfrac{6!}{3!(6-3)!}\cdot (0.5)^3\cdot (0.5)^3=20\cdot \dfrac{1}{2^6}=\dfrac{5}{16}[/tex]

2. 4 successes:

[tex]Pr_2=C^4_6p^4q^{6-4}=\dfrac{6!}{4!(6-4)!}\cdot (0.5)^4\cdot (0.5)^2=15\cdot \dfrac{1}{2^6}=\dfrac{15}{64}[/tex]

3. 5 successes:

[tex]Pr_3=C^5_6p^5q^{6-5}=\dfrac{6!}{5!(6-5)!}\cdot (0.5)^5\cdot (0.5)^1=6\cdot \dfrac{1}{2^6}=\dfrac{3}{32}[/tex]

4. 6 successes:

[tex]Pr_4=C^6_6p^6q^{6-6}=\dfrac{6!}{6!(6-6)!}\cdot (0.5)^6\cdot (0.5)^1=1\cdot \dfrac{1}{2^6}=\dfrac{1}{64}[/tex]

Now, the probability of at least three successes in six trials of a binomial experiment is

[tex]Pr=Pr_1+Pr_2+Pr_3+Pr_4=\dfrac{5}{16}+\dfrac{15}{64}+\dfrac{3}{32}+\dfrac{1}{64}=\dfrac{20+15+6+1}{64}=\dfrac{42}{64}=\dfrac{21}{32}=0.65625[/tex]

To find the probability of at least three successes in six trials of a binomial experiment where the success rate is 50%, we'll need to consider the complement of this event, which is easier to calculate in this situation. The complement consists of the probability of either 0, 1, or 2 successes in the six trials. By finding the sum of these probabilities, we can subtract it from 1 to find the probability of the original event (3 or more successes).

First, let's recall the formula for the binomial distribution:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

where:
- P(X = k) is the probability of k successes in n trials,
- C(n, k) is the number of combinations of n items taken k at a time, it can be calculated using the formula C(n, k) = n! / (k! * (n - k)!),
- p is the probability of success for each trial,
- (1 - p) is the probability of failure for each trial,
- n is the number of trials, and
- k is the number of successes.

Since the success probability is 50%, or 0.5, and the complement includes the probability of 0, 1, or 2 successes, we can calculate each of these probabilities.

For k = 0 (zero successes):
P(X = 0) = C(6, 0) * (0.5)^0 * (0.5)^(6 - 0)
P(X = 0) = (6! / (0! * 6!)) * 1 * (0.5)^6
P(X = 0) = 1 * (0.5)^6
P(X = 0) = (1/64)

For k = 1 (one success):
P(X = 1) = C(6, 1) * (0.5)^1 * (0.5)^(6 - 1)
P(X = 1) = (6! / (1! * 5!)) * (0.5) * (0.5)^5
P(X = 1) = 6 * (0.5) * (0.5)^5
P(X = 1) = 6 * (1/64)

For k = 2 (two successes):
P(X = 2) = C(6, 2) * (0.5)^2 * (0.5)^(6 - 2)
P(X = 2) = (6! / (2! * 4!)) * (0.5)^2 * (0.5)^4
P(X = 2) = (15) * (0.25) * (0.0625)
P(X = 2) = 15 * (1/64)

Now we sum up these probabilities to get the complement:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
P(X < 3) = (1/64) + 6*(1/64) + 15*(1/64)
P(X < 3) = (1 + 6 + 15) / 64
P(X < 3) = 22 / 64
P(X < 3) = 11 / 32

Now to find the probability of at least three successes (P(X >= 3)), we subtract the complement from 1:
P(X ≥ 3) = 1 - P(X < 3)
P(X ≥ 3) = 1 - (11 / 32)
P(X ≥ 3) = (32 / 32) - (11 / 32)
P(X ≥ 3) = 21 / 32

Converting this to a percentage and rounding to the nearest tenth of a percent:
P(X ≥ 3) ≈ (21 / 32) * 100
P(X ≥ 3) ≈ 65.625%

Rounded to the nearest tenth of a percent, the probability is 65.6%.

Answer:

c.) 18cm²

d.) 609m²

Step-by-step explanation:

I assumed that the question is the area of the grey fields.

c.) The white triangle is proportional to the big triangle whose side are known, (6,8). Thus the white trianlge's sides are (3,4). The area of a triangle is base*height / 2. Calculating the big triangle area subtracting the white triangle area gives the area of the grey. (6*8)/2-(3*4)/2 = 24 - 6 = 18.

d.) Area of a square is multiply the two sides. Adding together the greys, subtracting the whites will give the area of the grey. Small grey (11*14) Big grey (36*17) Big white (6*18) small white (7*7) -> 154+612-108-49=609.

The missing sides of the squares can be calculated by the given sides. Small grey 36-22=14. Big grey 28-11=17. Small white 13-6=7.

Answer:

A. Perpendicular

Step-by-step explanation:

Lines are perpendicular if their slopes are opposite reciprocals of each other. Opposite, meaning if positive, the other slope is negative, and if negative, the other slope is positive. Reciprocal meaning, the number is flipped upside down, turning fractions into whole numbers and vice versa.

1/9  

-1/9

-9

The slopes are perpendicular

Basically just add them all together like normal addition except there is a z attached to each number:

43z + 15z + 7z + 5z + 46z + 14z

58z+ 7z + 5z + 46z + 14z

65z + 5z + 46z + 14z

70z + 46z + 14z

116z + 14z

130z

Hope this helped!

Answer:

130z

Step-by-step explanation:

43z + 15z + 7z + 5z + 46z + 14z

Since the all have z with a coefficient, they are all like terms

Factor out a z

(43 + 15 + 7 + 5 + 46 + 14 ​ )z

Then add all the coefficients together

(130)z

The total is 130z

Answer:

Step-by-step explanation:

R means Rosa

given:

R+b+c=36

R R c-2=b, (R+2)+2=c

b-2=R, R+2=b

.

R+b+c=36 substitute for b and c

R+(R+2)+((R+2)+2)=36 add like terms

3R+6=36 subtract 6

3R=30 divide by 3

R=10

.

check

R=10

b=12

c=14

10+12+14=36

36=36

Answer: 10

Step-by-step explanation:

The children are 10, 12, and 14. (:

Any number (except zero) raised to the power of zero is one:

[tex]11^0=1[/tex]

On the other hand, by definition, the square of a number is that number multiplied by itself:

[tex]11^2=11\cdot 11 = 121[/tex]

So, the two answer are not the same:

[tex]11^0\neq 11^2[/tex]

After all, the exponential function

[tex]y=11^x[/tex]

is injective, which means that, given [tex]x_1\neq x_2[/tex], we have

[tex]11^{x_1}\neq 11^{x_2}[/tex]

Answer:

f(x) = x² - 2x - 15 passes through (-3 , 0) , (0 , -15) , (5 , 0)

f(x) = -x² - 2x + 15 passes through (-2 , 15) , (-1 , 16) , (0 , 15)

f(x) = -x² + 2x - 15 passes through (0 , -15) , (1 , -14) , (2 , -15)

Step-by-step explanation:

* Lets explain how to solve this question

- To find the points whose graph passes through them substitute the

  x-coordinate in the function if the answer is the same with the

  y-coordinate of the point then the graph passes through this point

  lets do that

- Check the first set of points with the first function

# Pint (0 , -15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (0)² - 2(0) - 15 = -15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , -15)

# Pint (1 , -14)

∵ f(x) = x² - 2x - 15

∴ f(0) = (1)² - 2(1) - 15 = -16 ⇒ not same value of y-coordinate

∴ The graph of the function does not pass through point (1 , -14)

∴ The graph does not pass through this set of points

- Check the second set of points with the first function

# Pint (-2 , 15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (-2)² - 2(-2) - 15 = 4 + 4 - 15 -7 ⇒ not same value of y-coordinate

∴ The graph of the function does not pass through point (-2 , 15)

∴ The graph does not pass through this set of points

- Check the third set of points with the first function

# Pint (-3 , 0)

∵ f(x) = x² + 2x - 15

∴ f(0) = (-3)² - 2(-3) - 15 = 9 + 6  -15 = 0 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-3 , 0)

# Pint (0 , -15)

∵ f(x) = x² - 2x - 15

∴ f(0) = (0)² - 2(0) - 15 = -15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , -15)

# Pint (5 , 0)

∵ f(x) = x² + 2x - 15

∴ f(0) = (5)² - 2(5) - 15 = 25 - 10  -15 = 0 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (5 , 0)

∴ The graph passes through this set of points

* f(x) = x² - 2x - 15 passes through (-3 , 0) , (0 , -15) , (5 , 0)

- Check the first set of points with the second function

# Pint (0 , -15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(0)² - 2(0) + 15 = 15 ⇒ not same value of y-coordinate

∴ The graph of the function does not passes through point (0 , -15)

∴ The graph does not pass through this set of points

- Check the second set of points with the second function

# Pint (-2 , 15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(-2)² - 2(-2) + 15 = -4 + 4 + 15 = 15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-2 , 15)

# Pint (-1 , 16)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(-1)² - 2(-1) + 15 = -1 + 2 + 15 = 16 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (-1 , 16)

# Pint (0 , 15)

∵ f(x) = -x² - 2x + 15

∴ f(0) = -(0)² - 2(0) + 15 = 15 ⇒ same value of y-coordinate

∴ The graph of the function passes through point (0 , 15)

∴ The graph passes through this set of points

* f(x) = -x² - 2x + 15 passes through (-2 , 15) , (-1 , 16) , (0 , 15)

- Now we have the first set of points and the third function

∴ The graph passes through this set of points

∴ f(x) = -x² + 2x - 15 passes through (0 , -15) , (1 , -14) , (2 , -15)

Answer:

y=72

x=0

Step-by-step explanation:

In the given question, Stuart is not correct.

The sum of all internal angles of a triangle is always equal to 180°.

Value of ∠1 = 65°(given)

Value of ∠7 = 55°(given)

Now let's look into Stuart's reasoning:

Step 1 : It is correct because according to angle sum property of triangle, ∠1 + ∠7 + ∠8 = 180°.

Step 2 : This step is wrong, because ∠4 and ∠8 are corresponding angles and not ∠12 and ∠8, so ∠4 = 60°.

Step 3 : This step is also wrong , ∠4 + ∠12 = 180°

             So ∠12 = 180 - 60 = 120°

Hence, Stuart's value of ∠12 is not correct.

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170 is the answer to this

Answer:

17,000.

Step-by-step explanation:

These are all of the steps to completely and correctly solve this question.

Hope this helps!!!

Kyle.

Answer:

B) As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞.

Step-by-step explanation:

Answer:

See attachment.

Step-by-step explanation:

The parent function is [tex]f(x)=|x|[/tex]

When this function is translated 2 units to the right, the new equation becomes; [tex]g(x)=|x-2|[/tex].

Another translation of 2 units up gives  [tex]h(x)=|x-2|+2[/tex].

A final dilation by a factor of [tex]\frac{1}{3}[/tex] gives  [tex]i(x)=\frac{1}{3}|x-2|+2[/tex].

The graph of this function is shown in the attachment.

Answer:

Its C

Step-by-step explanation:

On Edge

Let's translate the sentences into equations:

One number is 20 more than another: [tex]x=20+y[/tex]

If the greater number is increased by 4: [tex]x+4\ldots[/tex]

The result is five times the smaller [/tex]\ldots=5y[/tex]

So, we have the following system:

[tex]\begin{cases}x=20y\\x+4=5y\end{cases}[/tex]

Use the expression for x given by the first equation to solve the second:

[tex]x+4=5y \iff 20+y+4 = 5y \iff 24= 4y \iff y = 6[/tex]

which easily implies

[tex]x=y+20 = 26[/tex]

y- -8=(5/2)(x-4)

so is y+8=(5/2)(x-4)

Answer:

-12

Step-by-step explanation:

think of it like this

Do -48/2 which equals -24 then cut that in half which equals -12 and thats your answer

First add -2 and -6 together

Since these are both negitive signs you will add normally and and a negitive sign to the answer

-8

so you have...

-8 + 7

Since there is a negative (-8) and a positive (7) you will treat this as a normal subtraction problem, except your answer will have the sign of the biggest number

8 - 7 = 1

8 is the bigger number and has a negative sign therefore the answer is a negative number

so...

-8 + 7 = -1

-1

Hope this helped!

~Just a girl in love with Shawn Mendes

The answer to the rotation of triangle def is B)

Answer:

Step-by-step explanation:

To create triangle D'E'F', you need to rotate triangle DEF 180°, which results a figure with an opposite position, like a mirror. A 180° rotation always gives an opposite position, a mirror effect.

Answer:

y = (x+ 3)^2 - 10.

Step-by-step explanation:

Vertex form is

y =  a(x - b)^2 + c

Here b = -3 and c = -10 so we have

y = a(x + 3)^2 - 10   where a is some constant.

The y intercept is (0, -1)  so substituting:

-1 = a * 3^2 - 10

-1 + 10 = 9a

9a = 9

a = 1

So the required parabola is (x+ 3)^2 - 10.

Answer:

B

Step-by-step explanation:

Picture shows the formula to find the circumference of a circle

The circumference of a circle is calculated by multiplying the diameter by the mathematical constant π, typically approximated as 3.14159. This relationship is captured by the formula C = πd, which is fundamental in the study of geometry.

The relationship between the circumference and the diameter of a circle is described by the mathematical constant π (pi). The circumference (C) of a circle can be calculated by multiplying the diameter (d) of the circle by π. Therefore, the correct equation is b. Pi times the diameter equals the circumference, which can be expressed as C = πd.

This relationship is a fundamental aspect of Euclidean geometry. It is interesting to note that the diameter of the circle is twice the radius (d = 2r), so the circumference can also be calculated by the formula C = 2πr, where r is the radius. This equation highlights that the circumference is proportional to the diameter, with π serving as the constant of proportionality.

The simplified expression for cos x csc x tan x is 1 .

Sure, let's simplify the expression step by step:

Given expression:[tex]\( \cos(x) \csc(x) \tan(x) \)[/tex]

We know that:

[tex]- \( \csc(x) = \frac{1}{\sin(x)} \)[/tex]

[tex]- \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)[/tex]

So, we substitute these into the expression:

[tex]\( \cos(x) \cdot \frac{1}{\sin(x)} \cdot \frac{\sin(x)}{\cos(x)} \)[/tex]

Now, we cancel out the common terms:

[tex]\( \frac{\cos(x) \cdot \sin(x)}{\sin(x) \cdot \cos(x)} \)[/tex]

Now, we can see that the numerator and the denominator cancel each other out:

[tex]\( \frac{1}{1} = \boxed{1} \)[/tex]

In conclusion, the simplified expression is ( 1 ).

We start by using the trigonometric identities to express [tex]\( \csc(x) \) and \( \tan(x) \) in terms of \( \sin(x) \) and \( \cos(x) \)[/tex]. Then, we substitute these expressions into the given expression. Next, we cancel out the common terms in the numerator and denominator, resulting in a simplified expression of 1. This simplification demonstrates the relationship between the trigonometric functions and highlights their interconnectedness through fundamental trigonometric identities.

Complete question:

Simplify the expression cos x csc x tan x

- The perimeter for it is 8+V10 Units.

The perimeter of the triangle LMN is 8 + √10 units.

Perimeter of a straight sided figures or objects is the total length of it's boundary.

Given is a triangle LMN in the coordinate plane.

The coordinates of the vertices are L(2, 4), M(-2, 1) and N(-1, 4).

We have to find the length of each sides.

Using the distance formula,

LM = [tex]\sqrt{(-2-2)^2+(1-4)^2}[/tex] = √(16 + 9) = √25 = 5

MN = [tex]\sqrt{(-1--2)^2+(4-1)^2}[/tex] = √10

LN = [tex]\sqrt{(-1-2)^2+(4-4)^2}[/tex] = √9 = 3

Perimeter = LM + MN + LN = 8 + √10

Hence the perimeter is 8 + √10 units.

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Your question is incomplete. The complete question is as given below.

Answer:

42 feet

Step-by-step explanation:

This is a ratio question. The ratio of the height of the lamp to its shadow is the same as the ratio of the height of the building to its shadow.

So, 15/9 = 70/x, where x is the building's shadow.

Cross mutiply and solve for x.

15x = 630

x=42

Using the concept of similar triangles, we set up a proportion comparing the heights and shadow lengths of the lamp and building. After solving the proportion 15/9 = 70/x, we find that the shadow cast by the 70-foot tall building is 42 feet long.

To find the length of the shadow cast by the building, we can use the concept of similar triangles. The lamp and its shadow form one triangle, and the building and its shadow form a second triangle. These two triangles are similar because the angles are the same, meaning they have the same shape but are of different sizes.

Given that a 15-foot lamp casts a 9-ft shadow, we can set up the following proportion:
Lamp Height / Lamp Shadow = Building Height / Building Shadow, which simplifies to 15/9 = 70/x, where x is the length of the building's shadow we are trying to find.

By cross-multiplying, we get 15x = 9 * 70, which simplifies to 15x = 630. Dividing both sides by 15 gives us x = 42, so the shadow cast by the building is 42 feet long.

Answer:

138 cm.

Step-by-step explanation:

So first, we find the S.A. of the front and back.

The diagram says the side length of the front is 3 cm. and 3 cm.

3x3=9. So then, the back is also 9 cm, 9+9=18.

Now to find the S.A.'s of the four sides, you have to see the side lengths of each of them. The side lengths are 3 and 10.

3x10=30. This means each of them is 30 cm.

30x4=120. 120 is the total surface area of the four sides.

To find the total surface area of the whole rectangle, you add all the surface areas.

120+18=138 cm. (Not squared, since it's surface area and not area.)

ANSWER

16

EXPLANATION

The given expression is;

[tex]3 {x}^{2} + 4 {y}^{2} [/tex]

If x=2, y=1 and z=-3, we substitute the values into the expression and solve.

We substitute to obtain;

[tex]3 {(2)}^{2} + 4 {(1)}^{2} [/tex]

We evaluate to get;

[tex]3 {(4)} + 4 {(1)}[/tex]

We multiply out to get:

[tex]12+ 4 = 16[/tex]

Therefore the value of the given expression is with the given values is 16

Answer:

= -3x2 - 11x + 13

Step-by-step explanation:

=( +9) + (-3x2 - 11x + 4)

= -3x2 - 11x + 4 + 9

= -3x2 - 11x + 13

Answer:

11.77

Step-by-step explanation:

Volume of sphere = [tex]\frac{4}{3}[/tex] × π × r²

580 mm³ =  [tex]\frac{4}{3}[/tex] × π × r²

( Divide both sides by  [tex]\frac{4}{3}[/tex] )

435 mm³ = π × r²

( Divide both sides by π )

138.4648005 = r²

( Square root both sides )

11.76710672 = r

Answer:

pift^2(or your third option) is the area of a circle with a radius of 1.

Answer:  -5/21

Step-by-step explanation:

-3/7 & -4/6

common detonator is 42

-3/7 = -18/42 reduced to -9/21

-4/6 = -28/42 reduced to -14/21

difference between -9/21 and -14/21 = -5/21