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10: Find the x-and y-intercepts of the graph of the function f(x) = x^4 − 1
Solution :

Therefore , the solution of the given problem of equation comes out to be the vertex (h,k) is  (3,3).

An expression is a mathematical representation of two equal variables , one on each end of a "equals" sign. Typical issues can be resolved using equations. We commonly look to pre algebra for solutions to problems in real life. The fundamental concepts of mathematics are covered in pre-algebra lessons.

Here,

use the vertex form y=a(x-h)2+k to get the values of a, h, and k.

Given : a =1

h=3

k=−3

Discover the vertex (h,k) (3,3).

So, we discovered the vertex of given equation.

Therefore , the solution of the given problem of equation comes out to be the vertex (h,k) is  (3,3).

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Answer:

Step-by-step explanation:

The table that represents a linear function is ob x 0 1 2 3 y 0149

On solving the provided question, we can say that from the graphs the function = [tex]f(x) = 1/2 (3)^x[/tex], it represents a half parabola

The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope. Usually, f is used to denote functions (x). input is an x. There are four main types of functions accessible. based on the following factors: on functions, one-to-one functions, many-to-one functions, inside functions, and on functions.

from the graphs

the function = [tex]f(x) = 1/2 (3)^x[/tex]

it represents a half parabola

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The greatest error that can be tolerated in the measurement of expressed as [tex]\frac{1}{3}%h[/tex]%h.

The given data is the height and radius of a right circular cylinder are equal.

[tex]$$\begin{aligned}V & =\pi h^3 \\\frac{d V}{d h} & =3 \pi h^2 \\d V & =3 \pi h^2 d h\end{aligned}$$[/tex]

A differentiable function is a function in one variable in calculus such that its derivative exists at each point in its entire domain. The tangent line to the graph of a differentiable function is always non-vertical at each interior point in its domain.

A differentiable function does not have any break, cusp, or angle.

Find the derivative of V in terms of h. Then multiply both sides by dh.

The maximum percentage error of the volume is 1%.

[tex]$$\begin{aligned}0.01 \pi h^3 & =3 \pi h^2 d h \\\frac{0.01}{3} h & =d h \\\frac{1}{300} h & =d h\end{aligned}$$[/tex]

The maximum percentage error of the volume is 1% which means

dV=0.01=0.01[tex]\pi[/tex][tex]h^{3}[/tex]. Substitute this into the equation and then solve for

dh.

The maximum error in the height is then [tex]\frac{1}{300}h=\frac{1}{3}[/tex]%h.

Therefore, the greatest error that can be tolerated in the measurement of expressed as [tex]\frac{1}{3}%h[/tex]%h

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The curvature of r(t) at the point (3,1,1) is 2/3

The curvature of a space curve is given by the formula:

k(t) = |r'(t) x r''(t)| / |r'(t)|^3 where,

r(t) = (3t, t^2, t^3)

r'(t) = (3, 2t, 3t^2)

r''(t) = (0, 2, 6t)

So we can substitute these in the above formula:

k(t) = |(3, 2t, 3t^2) x (0, 2, 6t)| / |(3, 2t, 3t^2)|^3

By cross-product, we get

k(t) = |(6t, -6, -6t^2)| / |(3, 2t, 3t^2)|^3

Now substituting the point (3,1,1) we get

k(t) = |(6(1), -6, -6(1)^2)| / |(3, 2(1), 3(1)^2)|^3 = 6/9 = 2/3

So the curvature of r(t) at the point (3,1,1) is 2/3

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Option A is appropriate since the table's function is linear.

The equals sign is used in equations to indicate the equality of two expressions in a formula. The expressions relate two or more variables with each other.

Given that:

x = 2 and y =4

x= 4 and y= 8 and so on.

In all the values given, the value of y is two times the value of x.

Also, these values satisfy the condition of linear expression:

y=mx

where, m =2 for the represented value.

This means that the function is linearly dependent on x.

The function shown in the table is linear.

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The probability of getting 2 reds will be 13.3%.

The probability of a red and a yellow will be 22.2%.

Probability simply means the chance that a particular thing or event will happen. It is the occurence of likely events. It is simply the area of mathematics that deals with the numerical estimates of the chance that an event will occur or that a particular statement is true

From the information the bag contains 1 green, 4 red, and 5 yellow balls. Two balls, are selected at random.

The probability of getting 2 reds will be:

= 4/10 × 3/9

= 12/90

= 13.3%

The probability of a red and a yellow will be:

= 4/10 × 5/9

= 20/90

= 22.2%

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The required sum of the 7 terms in the geometric sequence is 16.6.

Geometric progression is a sequence of series whose ratio with adjacent values remains the same.

Here,
Given geometric series,
10, 4, 8/5 . . . . . .  

Common ratio = 4 / 10 = 2/5

first term = 10

the sum of the first seven terms is given as,
S₇ = a[r⁶ - 1] / r - 1
S₇ = 10[{2/5}⁶ - 1] / {2/5 - 1}
S₇ = 16.6

Thus, the required sum of the 7 terms in the geometric sequence is 16.6.

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The quotient of the given answer is 0.

The divisor is the integer that divides a given number. The quotient is the sum that we arrive at as a result. The residual is the number that the divisor leaves after partially dividing the original integer.

Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English.

On dividing 49 by 679 we will get quotient 0 and remainder 49

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The interval over which h(t) has a negative rate of change is given as follows:

(-∞, -3).

The equation of a quadratic function, of vertex (h,k), is given by the following rule:

y = a(x - h)² + k

In which:

The equation for this problem is given as follows:

h(t) = (t + 3)² + 5.

The leading coefficient is positive, hence the function is concave up.

Considering the t-coordinate of t = -3 at the vertex, the decreasing and increasing intervals are given as follows:

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The given two lines are parallel to each other. The correct answer would be an option (A).

The slope of the line is defined as the gradient of the line. It is denoted by m

Slope m = (y₂ - y₁)/(x₂ -x₁ )

The lines will be parallel if they have the same slope and they will be perpendicular if the product of the slopes of the two lines is -1.

The slope of the first line can be found using the formula:

m₁ = (-2 - (-5))/(5-9) = -3/4

The slope of the second line can be found using the formula:

m₂ = (1 - (-2))/ (-8 - (-4)) = -3/4

They have identical slope, so the lines will be parallel.

Hence, the given two lines are parallel to each other.

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Option A is the dot plots which would most likely have the highest standard

deviation.

This is referred to as the measure of the amount of variation or dispersion of a set of values and the one with a higher standard deviation will usually have its average points farther away from the mean which is located at the center while those with a lower  standard deviation will usually have its average points closer to the mean.

From the options provided , option A has a lot of points on 2 and 14 which are further way from the center (mean) and is therefore the reason why it was chosen as the correct choice.

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The lower bound on the probability that the plant will fill between 9,000 and 11,000 cans on a particular day is at least 1 - 1/10 = 9/10.

What is probability ?

A probability is a number that reflects the chance or likelihood that a particular event will occur.

(a) P(X > 11,000) ≤ E[X] / 11,000

Where X is the random variable representing the number of cans filled in a day, and E[X] = 1,000 is the expected value.

So, P(X > 11,000) ≤ 1,000 / 11,000 = 1/11

(b) P(|X - E[X]| ≥ kσ) ≤ 1/k^2

Where X is the random variable representing the number of cans filled in a day, E[X] = 1,000 is the expected value, and σ is the standard

As we know that for any random variable X, E[X^2] >= (E[X])^2

therefore Var(X) = E[X^2] - (E[X])^2 >= 0.

so the standard deviation of X is

σ = √(Var(X)) = √(1000) = 31.62

We can choose k = (11000-9000)/(2*31.62) = 3.16

so P(|X - E[X]| ≥ 3.16σ) = P(|X - 1000| ≥ 10000) ≤ 1/3.16^2= 1/10

Therefore, the lower bound on the probability that the plant will fill between 9,000 and 11,000 cans on a particular day is at least 1 - 1/10 = 9/10.

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Variance and standard deviation are two common measures of spread in statistics. The standard deviation is √30.25 = 5.5.

OB. s² = The sample variance is a measure of the spread of a set of data. To calculate the sample variance, the sum of the squares of the differences between each data point and the mean is divided by the number of data points minus one. In this case, the mean of the data set is (22+15+4+7+9)/5 = 11.

The differences between each data point and the mean are 11-22=-11, 11-15=-4, 11-4=7, 11-7=4, and 11-9=2. The sum of the squares of these differences is (-11)² + (-4)² + 7² + 4² + 2² = 121.

Dividing this by the number of data points minus one, which is 5-1=4, gives us a sample variance of 121/4 = 30.25.

The standard deviation is the square root of the sample variance, so in this case, the standard deviation is √30.25 = 5.5.

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Answer:

[tex]y = \frac{1}{10}x - 1[/tex]

Step-by-step explanation:

We are given the equation x - 10y = 10.

We want to write it in slope-intercept form.

Slope-intercept form is given as y=mx+b, where m is the slope and b is the value of y at the y intercept, hence the name.

Notice how in slope-intercept form, only y is on one side.

This is because the equation is basically solved for y in slope-intercept form.

So, we should solve the equation for y.

Start by subtracting x from both sides.

x - 10y = 10

-x           -x

_______________

-10y = -x + 10

Now, divide both sides by -10 to isolate y.

-10y = -x + 10

÷10    ÷10

_______________

[tex]y = \frac{1}{10}x - 1[/tex]

Answer:

We can start solving this problem by using a system of linear equations. Let x be the number of gallons of the 20% pure slime mixture and y be the number of gallons of the 80% pure slime mixture. We know the following:

The total amount of slime mixture is 1000 gallons. This means that x + y = 1000.

The final mixture is 65% pure slime, which means that 0.65 = (0.20)x + (0.80)y. This is because we are mixing 20% pure slime and 80% pure slime to get the final mixture.

We can use these two equations to solve for x, which is the number of gallons of the 20% pure slime mixture needed.

First, we'll use the first equation to solve for y in terms of x:

y = 1000 - x

Then we'll substitute this expression into the second equation:

0.65 = (0.20)x + (0.80)(1000 - x)

Simplifying we get:

0.65 = 0.20x + 800 - 0.80x

0.45x = 200

x = 200/0.45

x = 444.44 gallons

So, you would need 444.44 gallons of the 20% pure slime mixture to make 1000 gallons of a 65% pure slime mixture.

Step-by-step explanation:

The value of x, which is the measure of the remaining angle, is equal to 180° - 70° - 30°, or 80°.

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Answer: 400

Step-by-step explanation: Max has rounded off the difference of two numbers. Max's estimate of the difference is 400.

Given information:

The first number = 718.

The second number = 331

Now, according to the question, Max estimates their difference by taking the number rounded off.

Hence, the rounding off the given number to nearest 100,

we get,

First number as 700 and Second number as 300.

Hence the difference of the obtained number can be written as:

Hence, one can conclude that, max's estimate of the difference is 400.

The volume, V of box as a function of x that is V(x) = 4x³ - 64x² + 240x , where x is side of square cut out from each corner.

We have, a box open from top and constituted from rectugular pieces of dimensions 12 inches and 20 inches. As we know, The volume of a box is V

= width (w) × height(h) ×length (l).

Now, equal squares of side x are cut from each corner of box and folded. We draw our box it looks like as seen above figure. We will notice here that the height will be x and that the height and length depend on x. Length of box, l = 20 - 2x and

Width of box, w = 12 - 2x

So the volume is, V(x) = w×h×l

= (12 - 2x)(x)(20 - 2x)

Expand it for further calculation,

V(x) = ( 12 - 2x ) ( 20x - 2x²)

V(x) = 4x³ - 64x² + 240x

Hence, the required volume function is V(x) = 4x³ - 64x² + 240x.

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The volume, V, of a box is defined as V(x) = 4x3 - 64x2 + 240x.

We have a box that opens from the top and is made of rectangular components that are 12 inches and 20 inches in length. As is common knowledge, a box's volume is V.

V = height (h) + width (w) + length (l).

From each corner of the box, equal squares of side x are now cut and folded. We create a box that resembles the image above. Here, we'll see that the height is x and that the relationship between height and length is x-dependent.

Box length, l = 20 - 2x, and

Box width, w = 12 - 2x

Thus, V is the volume (x) = W * H * I

V(x) = (12 - 2x) (x) (20 - 2x)

V(x) = ( 12 - 2x ) ( 20x - 2x²)

V(x) = 240x - 24x² - 40x² -4x³

V(x) = 4x³ - 64x² + 240x

Hence, the required volume function is V(x) = 4x³ - 64x² + 240x.

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A graph which represents the system of inequalities for this scenario include the following: graph B.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of developed acres and number of open acres, and then translate the word problem into algebraic equation as follows:

Since a planned community must have at least 4 acres of developed and open space, a linear inequality which represents this situation is given by;

x + y ≥ 4

Additionally, since the difference between the number of developed acres and the number of open acres can be no more than 1, we have;

y – x ≤ 1

Next, we would use an online graphing calculator to graphically solve the system of linear inequalities. Therefore, the required solution for the given system of linear inequalities is the shaded area below the solid line, which is given by graph B.

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Changing the compound inequality x > −3 and x < 3 from “and” to “or” will change the solution set

A sentence having two inequality statements connected by the words "or" or "and" is referred to as a compound inequality.

The conjunction "and" denotes the simultaneous truth of both statements in the compound sentence. It is when the solution sets for the various statements cross over or intersect.

The conjunction "or" means that the entire compound sentence is true as long as either statement is true. When "or" appears in a compound inequality, a disjunction is created.

Hence changing “and” to “or” changes the solution set

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Answer:

7/10

Step-by-step explanation:

steps linked below

Answer: y= -3x+7

Step-by-step explanation:

the equation is y=Mx+b you gave me the slope (-3) and the y-intercept (7) so the answer is y=-3x+7

Answer:

-1

Step-by-step explanation:

Since -1 to a power will either be negative or postive 1, looking at whether the exponent is odd or even can determine if the answer is -1 or 0 (x + x^2 for instance  equals 0 but x + x^2 + x^3 is -1)

[tex]\it (-1)^{2k}=1;\ \ \ (-1)^{2k+1}=-1,\ \ \forall\ k \in\mathbb{N}\\ \\ S=-1+(1-1)+(1-1)+\ ...\ (1-1)=-1[/tex]

Answer:

$63

Step-by-step explanation:

12-5=7

9*7= 63

Answer:

$63 Is the answer.

Step-by-step explanation:

$12 - $5 = $7
$7 x $9 = $63

Answer:

Out of all possible choices you've shown, I'd say the best answer would be D

Step-by-step explanation:

The truth table for  ~p ∨ q is as shown below.

~p ∨ q

   T

   F

   T

   T

A truth table is a table for evaluating the truth value of a truth-functional compound sentence on the basis of the truth value of its part

The truth tables for negation and disjunction are shown below:

Negation

p     ......    ~p

T                F

F                T

Disjunction

p  ..... q  ..... p v q

T        T         T

T        F         T

F        T         T

F         F        T

Note:

T and F stand for True and False respectively.

In disjunction, if one of the inputs is true then the output will be True.

Negation reverses the input i.e if the input is True, the output will be False. The symbol for negation is  ~ i.e the negation of p is  ~p

Truth table for ~p ∨ q?

p  -----  q  -----  ~p  -----  q  -----  ~p ∨ q

T          T            F         T              T  

T          F            F         F               F

F          T            T         T              T    

F          F            T         F              T  

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Answer:

as: 4/3 * (2 * 1/3)

Step-by-step explanation:

Note: parentheses, 2 and 1/3 are being multiplied together first, before being multiplied by 4/3.

Therefore , the equation problem is  15 fraction of 72 is the number 480. The expressions published here on left & right sides appear to be equivalent in their relationship.

Mathematical statements with two algebraic in the middle and also the means total (=) sign from either side are called equations. Variables, contractors, diffusion coefficients, terms, and the equatable to sign are just a few of the elements that compose an equation. The "=" sign but also terms both on sides are always necessary when writing an equation.

Here,

Using the value A = 480 as a guide

Let y become the ratio of the number, or the opposite.

It is equivalent to 72%.

The equation that results is

72 is equivalent to x% of 480.

When the expression is condensed, we get

72 = y % x 480

72 = ( y/100 ) x 480

Divide each side by 480 to obtain

y/100 = 0.15

Multiply by 100 on each side to arrive at

y = 15

Consequently, the value is y = 15.

Therefore, out of 480, 72 represents 15%.

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