4.1 Find the intervals on which the function = [tex]x^{3}[/tex] −
4[tex]x^{2}[/tex]$ + 10 is increasing or decreasing.

The probability the point will be in the region labeled A is 2/15 if  point is selected at random inside the given figure.

Given that a point is selected at random from inside the given figure.

We are to find the probability that the point will be in the region labeled B.

From the figure, we note that the regions A, B and D are rectangles and the region C is a square.

The areas of all the regions are calculated as follows:

Area of region A is 5×(3+4) =35 sq in

Area of region B is 3×4  = 12 in

Area of region C is 4² = 16 sq in

Area of region D is 3×(4+5)= 27 sq. in

Therefore, the probability that the randomly chosen point will lie in the region B is given by P

P = 12/35+12+16+27

=12/90

=2/15

Hence,  the probability the point will be in the region labeled A is 2/15

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The diameter of the wheel is approximately 45 cm.

To find the diameter of the wheel, we need to determine the circumference of the wheel using the number of revolutions and the distance traveled. The formula for the circumference of a wheel is given by:

Circumference = 2πr

where r is the radius of the wheel. Since we need to find the diameter, we can substitute r with d/2, where d is the diameter. The formula becomes:

Circumference = πd

We are given that the wheel completes 25 revolutions and travels about 35.325 meters. The distance traveled in one revolution is equal to the circumference of the wheel. Let's calculate the circumference of the wheel first:

Circumference = 35.325 m / 25 = 1.413 m

Now we can equate this to πd and solve for d:

1.413 = πd

To find the diameter, we divide both sides of the equation by π:

d = 1.413 / π ≈ 0.449 m

Finally, to convert the diameter to centimeters, we multiply by 100:

d ≈ 0.449 m * 100 ≈ 44.9 cm

Rounding this value to the nearest centimeter, the diameter of the wheel is approximately 45 cm.

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Answer: The total cost will be equal to $2839.16. The expression to calculate the total cost is 119.92(m)+81=C(t).

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that a person buys a phone for ​$81 and signs up for a​ single-line phone plan with 2000 monthly anytime minutes. The plan costs ​$119.92 per month.

The expression will be formed as below:-

119.92(m)+81=C(t)

119.92(23)+81=C(t)

2839.16=C(t)

Therefore, the total cost will be equal to $2839.16.

If owners of a recreation area are filling a small pond with water.  The  total amount of water after 13 minutes is 1016 liters.

Let the total amount of water in the pond = W

Let the total number of minutes that water has been added = T

The equation relating W to T Is:

W = 600 + 32T

Where:

600=  initial amount of water in the pond

32T = additional water added over time

Now let find the  total amount of water in the pond after 13 minutes

Substitute T = 13 into the equation:

W = 600 + 32(13)

W = 600 + 416

W = 1016

Therefore the total amount of water is 1016 liters.

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The perimeter of this solid is 72 cm.

In Mathematics and Geometry, the perimeter of a triangle can be calculated by using this mathematical equation:

P = a + b + c

Where:

Since the shape was formed by combining four (4) of these triangles, we can logically deduce the following side lengths;

Side length = 12 - 9 = 15 - 12

Side length = 3 cm.

Therefore, the perimeter of this solid can be calculated as follows;

Perimeter of solid = 4(15) + 4(3)

Perimeter of solid = 60 + 12

Perimeter of solid = 72 cm.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

We find that Mary owes $13,564 in income tax. So the correct option is (a) $13,564. Option a

Mary's taxable income falls between $31,850 and $77,100. This means we apply the tax rate relevant to this bracket, which is $4,386.25 plus 25% of the amount over $31,850.

First, we need to subtract $31,850 from Mary's income to find the amount that we need to apply the 25% tax rate to:

$68562 - $31850 = $36712

Next, we take 25% of this result:

0.25 * $36712 = $9178

Then we add the base tax for this bracket ($4,386.25) to this result to get the total tax:

$4386.25 + $9178 = $13564.25

Rounding this to the nearest dollar, we find that Mary owes $13,564 in income tax. So the correct option is (a) $13,564.

Option  a

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We can conclude that cosθ = sin(90° - θ) for any right triangle with angle θ and its complementary angle (90° - θ).

To prove that cosθ = sin(90° - θ), we can use the properties of right triangles and their angles.
Consider a right triangle ABC, where angle A is θ, angle B is (90° - θ), and angle C is 90°.
According to the definition of trigonometric functions in a right triangle:
- cosθ = adjacent side (AB) / hypotenuse (AC)
- sin(90° - θ) = opposite side (AB) / hypotenuse (AC)
From the given definitions, we can see that both cosθ and sin(90° - θ) have the same ratio of side lengths, which is AB/AC.
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The third point so the hypotenuse of the triangle has a length of 13 could be; (-4, 2)

We are given 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.

|[tex]AC|^2 = |AB|^2 + |BC|^2[/tex]

we know that 5, 12, 13 is a Pythagorean triple;

[tex]c^2-a^2=b^2[/tex]

[tex]13^2-5^2=b^2[/tex]

[tex]169-25=b^2[/tex]

[tex]144=b^2[/tex]

12=b

The point so the hypotenuse of the triangle has a length of 13 is (-4, 2)

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The  circle is:[tex]x^2+y^2 = 5^2 = 25[/tex]This circle has center (0,0) and radius 5 units. Therefore, it is a circle with a radius of 5 units, and every point that lies on the circumference of this circle is 5 units away from the center of the circle.

the distance between any point on the circle and the center of the circle is always 5 units. This is because the equation of a circle is

[tex](x-a)^2+(y-b)^2=r^2[/tex]

where the center is (a,b) and the radius is r units. In this case,

a=b=0 and r=5.

Algebraically show that the point (6, 8) lies on the circle.

To show that the point (6, 8) lies on the circle, we need to substitute the values of x and y in the equation of the circle and verify whether the equation holds true or not.So,

substituting x=6 and y=8 in the equation of the circle, we

[tex]get:6^2 + 8^2 = 36 + 64 = 100[/tex]

Now, we can see that the LHS of the equation is equal to 100, which is also equal to the square of the radius of the circle. Therefore, the point (6, 8) lies on the circumference of the circle with center (0,0) and radius 5 units.

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The residual measures the difference between the predicted and observed dosage, indicating if the customer received more or less medication than expected based on their mass.

The residual in this context refers to the difference between the predicted dosage of the medicine, based on the least-squares regression line, and the actual dosage observed for a specific customer. It indicates how much the actual dosage deviates from the expected dosage based on the customer's mass.

In the scatter plot, the residual for a particular customer represents the vertical distance between the observed data point (the actual dosage) and the regression line (the predicted dosage).

If the residual is positive, it means the observed dosage is higher than the predicted dosage, indicating that the customer received a higher dosage of the medicine than expected based on their mass.

Conversely, if the residual is negative, it means the observed dosage is lower than the predicted dosage, indicating that the customer received a lower dosage than expected based on their mass.

Interpreting the residual requires considering the magnitude of the deviation. A larger residual suggests a greater difference between the observed and predicted dosages, indicating a potentially significant variation from the regression model.

A smaller residual indicates a closer alignment between the observed and predicted dosages, suggesting that the regression model provides a more accurate estimation.

Overall, analyzing the residuals helps assess the effectiveness of the regression model in predicting the dosage of the medicine based on customer mass and identifies any outliers or discrepancies in the data that may need further investigation.

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g(x) shifts the graph vertically upward by 4 units,h(x) shifts the graph horizontally to the left by 7 units,k(x) scales the graph vertically by a factor of 20 and shifts it vertically upward by 10 units and m(x) compresses the graph horizontally by a factor of 1/4 and shifts it horizontally to the right by 1 unit.

To describe the graph of each function in terms of the graph of f(x) = 5 sin(x), we will analyze how each transformation affects the original graph.

1. g(x) = 5 sin(x) + 4:

Adding a constant value of 4 to the function f(x) shifts the graph vertically upward by 4 units. The amplitude and period of the graph remain the same.

2. h(x) = 5 sin(x + 7):

Adding a constant value of 7 inside the sine function causes a horizontal shift to the left by 7 units. This means the graph is shifted 7 units in the opposite direction of the sign (left in this case). The amplitude and period remain unchanged.

3. k(x) = 20 sin(x) + 10:

Multiplying the entire function by a constant of 20 scales the graph vertically, stretching it vertically by a factor of 20. Additionally, adding a constant value of 10 shifts the graph vertically upward by 10 units.

The amplitude remains the same, but the graph is vertically elongated and shifted upward.

4. m(x) = 5 sin(4x - 1):

Inside the sine function, the value of 4 stretches the graph horizontally by a factor of 1/4. This results in a shorter period, meaning the graph oscillates more frequently.

The value of -1 inside the sine function causes a horizontal shift to the right by 1 unit. Therefore, the graph is compressed horizontally and shifted 1 unit to the right. The amplitude remains the same.

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The product -2jk is negative if and only if either j or k is negative (but not both). Therefore, the possible conditions for j and k are:

j < 0 and k > 0

j > 0 and k < 0

For the expression -2jk to be negative, at least one of the factors j or k must be negative, since the product of two positive numbers is always positive, and the product of two negative numbers is positive as well. Therefore, the possible conditions for j and k are:

j is negative and k is positive.

j is positive and k is negative.

In other words, if j and k have opposite signs, their product will be negative. However, if both j and k are positive or both are negative, their product will be positive. It's important to note that there are infinite possibilities for j and k that satisfy these conditions, since j and k can take on any real value as long as one is negative and one is positive.

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The minimum number of times Johnny needs to use to the wheelbarrow until the delivery truck is filled up to 70%, whereby the shape of the truck is a rectangular prism is about 41 times

A rectangular prism is a three dimensional geometric figure with three pairs of parallel facing sides, and perpendicular adjacent faces.

Whereby the wheelbarrow and the truck are rectangular prisms, we get;

The dimensions of the wheel barrow are;

2 feet by 3 feet by 1.5 feet

The dimensions of the truck are;

11 feet by 8 feet by 6 feet

The amount of soil John puts in the truck = 70% of the volume of the truck

Therefore;

The amount of soil John puts in the truck = 11 feet × 8 feet × 6 feet × 70%

11 feet × 8 feet × 6 feet × 70% = 528 ft³ × 70% = 369.6 ft³

The number of times to use the wheelbarrow, n, is therefore;

n = 369.6 ft³ ÷ (2 ft × 3 ft × 1.5 ft) ≈ 41.0

The minimum number of times Johnny needs to use the wheelbarrow until the delivery truck is 70% filled is therefore 41  = 41 times

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Answer:

10

Step-by-step explanation:

the answer is in the question

The measure of angle U is given as follows:

m < U = 66º.

The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation is true to obtain the missing length:

c² = a² + b² - 2abcos(C)

Hence the length of side v is obtained as follows:

v² = 22² + 79² - 2 x 22 x 79 x cosine of 51 degrees

v² = 4537.48

[tex]v = \sqrt{4537.48}[/tex]

v = 67.36.

By the law of sines, we have that:

sin(51º)/67.36 = sin(U)/79

Hence the measure of angle U is obtained as follows:

sin(U) = 79 x sine of 51 degrees/67.36

sin(U) = 0.9114

m < U = arcsin(0.9114)

m < U = 66º.

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Answer:

False

Step-by-step explanation:

Net worth is a combination of everything you own.

This includes the amount of money you have, or have ever made, plus your house, vehicle, belongings, etc.

Answer:

Step-by-step explanation:

Let’s assume the original price of the item is “x.” Then, using the first statement, the list price of the item is 80% of the original price, or 0.8x. If there is a discount applied, let’s say “d,” then the discounted price would be (1-d)(0.8x).

Using the second statement, if the price of the item has been reduced by 80%, then the discounted price would be 0.2x. This can be expressed as (1 - 0.8)(x).

So, we have the following two equations:

(1 - d)(0.8x) = 0.2x

0.64x - 0.8dx = 0.2x

Simplifying this equation, we get:

0.44x = 0.8dx

d = 0.55

This means that the discount applied in the first statement is 55%, not 80%.

The calculation for change in percentage (increase or decrease) involves finding the difference between two values and expressing it as a percentage of the original value. This is different from multiplying by percentages, which involves finding a percentage of the original value and subtracting or adding it to the original value.

The wording of percentage problems is important because it can affect the way the problem is interpreted and the calculation that is used to solve it. For example, the phrase “increased by 50%” could be interpreted as multiplying the original value by 1.5, while the phrase “increased to 50%” could be interpreted as finding 50% of the original value and adding it to the original value.

Examples:

A shirt is on sale for 30% off its original price of $50. The discounted price is (1-0.3)($50) = $35.

A company’s revenue increased from $100,000 to $120,000. The percentage increase in revenue is ((120,000 - 100,000) / 100,000) x 100% = 20%.

The height of the can is approximately 23.33 cm.

We have,

The formula for the surface area of a cylinder.

S = 2πr² + 2πrh

where S is the surface area, r is the radius, h is the height, and π is pi (approximately 3.14).

In this case, we know that the diameter of the can is 6 cm, which means that the radius is half of that or 3 cm.

We also know that the surface area is 140 square centimeters.

So,

140 = 2π(3²) + 2π(3)(h)

Simplifying:

140 = 18π + 6πh

Dividing both sides by 6π:

23.333... = h

Rounding to the nearest hundredth:

h ≈ 23.33 cm

Therefore,

The height of the can is approximately 23.33 cm.

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The conversion of 2553 base 10 to base 7 is 10305 base 7.

We are given that;

The value= 2553 base 10

Now,

To convert 2553 base 10 to base 7, we need to divide 2553 by 7 repeatedly and write down the remainders. The final answer will be the remainders read from bottom to top. Here are the steps:

Divide 2553 by 7. The quotient is 364 and the remainder is 5. Write down 5.

Divide 364 by 7. The quotient is 52 and the remainder is 0. Write down 0 below 5.

Divide 52 by 7. The quotient is 7 and the remainder is 3. Write down 3 below 0.

Divide 7 by 7. The quotient is 1 and the remainder is 0. Write down 0 below 3.

Divide 1 by 7. The quotient is 0 and the remainder is 1. Write down 1 below 0.

Therefore, by conversion the answer will be 10305 base 7.

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2081 is the estimated population of the deer as per the given table.

We assume that the characteristics of the sample of the deer population match the characteristics of the population as a whole. This will be the case if the sample is random: each deer is equally likely to be included in the sample.

So, if 85 of the 110 marked deer show up in the sample, we assume that the sample is 85/110 of the entire population.

For a sample size of 1608 deer, that means the population is estimated to be ...

 1608 = 85/110 × population

 population = 1608 × 110/85 = 2081

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Jamal's Net Monthly Income (NMI) is $431.40.

We need to calculate his gross income and all the monthly deductions first.

Gross Income = Hourly Rate x Total Hours Worked = $8.75/hour x 63 hours = $551.25

Monthly Federal Income Tax = Gross Income x Federal Tax Rate = $551.25 x 0.10 = $55.125

Monthly Social Security (FICA) = Gross Income x FICA Rate = $551.25 x 0.062 = $34.13625

Monthly Medicare = Gross Income x Medicare Rate = $551.25 x 0.0145 = $7.987625

Monthly State Tax = Gross Income x State Tax Rate = $551.25 x 0.04 = $22.05

Monthly Local Tax = Gross Income x Local Tax Rate = $551.25 x 0.001 = $0.55125

Monthly Federal Income Tax + Monthly Social Security (FICA) + Monthly Medicare + Monthly State Tax + Monthly Local Tax = Total Monthly Deductions

= $55.125 + $34.13625 + $7.987625 + $22.05 + $0.55125

= $119.85

Net Monthly Income (NMI) = Gross Income - Total Monthly Deductions

= $551.25 - $119.85

= $431.40

Therefore, Jamal's Net Monthly Income (NMI) is $431.40.

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The solution to the given logarithmic expression is: x = 111.63

The logarithmic form of expression is written as [tex]log_{a} c = b[/tex]. This is simply a rearrangement of the specific exponential form, aᵇ = c. Thus, we can say that exponential equation could be written as a logarithm.

We want to solve the logarithmic expression given as:

log x = [tex]\frac{log 27}{log 5}[/tex]

log x = [tex]\frac{log 3^3}{log 5}[/tex]

log x = [tex]\frac{3 log 3}{log 5}[/tex]

log x = 2.0478

x = [tex]10^{2.0478}[/tex]

x = 111.63

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The equation of the line in the graph passing through the points (-2,1) and (6,7) is  [tex]y = \frac{3}{4}x + \frac{5}{2}[/tex].

The slope-intercept form is expressed as;

y = mx + b

Where m is slope and b is the y-intercept.

From the graph, the line passes through point (-2,1) and (6,7).

First, we determine the slope:

[tex]m = \frac{y_2 - y_1}{x_2 - x_1} \\\\m = \frac{7 - 1}{6-(-2)} \\\\m = \frac{6}{6+ 2} \\\\m = \frac{6}{8} \\\\m = \frac{3}{4}[/tex]

Now, plug the slope m = 3/4 and point (-2,1) into the point-slope form:

( y - y₁ ) = m( x - x₁ )

[tex]y - 1 = \frac{3}{4}(x - (-2)) \\\\y - 1 = \frac{3}{4}(x + 2)\\\\y - 1 = \frac{3}{4}x + \frac{3}{2} \\\\y = \frac{3}{4}x + \frac{3}{2} + 1\\\\y = \frac{3}{4}x + \frac{5}{2}[/tex]

Therefore, the euation of the line is [tex]y = \frac{3}{4}x + \frac{5}{2}[/tex].

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Answer:

[tex]\sf g(x) = f(x\; \boxed{+ 3}\;) \;\boxed{- 6}[/tex]

Step-by-step explanation:

A translation is a transformation that moves every point of a figure the same distance and in the same direction. This means that the size, shape, and orientation of the figure are preserved, but its position is changed.

From inspection of the given diagram, we can see that the size, shape and orientation of the graph of function g is the same as that of the graph of function f, but its position has changed.  Therefore, the transformation is a translation.

We can use the vertex of both graphs to determine the translation.

Therefore, the graph of function f has been moved 3 units left and 6 units down to create the graph of function g.

When we move "a" units left, we add the value of "a" to the x-value of the function.

When we move "a" units down, we subtract the value of "a" from the function.

Therefore:

[tex]\sf g(x) = f(x\; \boxed{+ 3}\;) \;\boxed{- 6}[/tex]

The y-intercept of the line [tex]y = -5x \ is \ 0[/tex].

The equation of a line in slope-intercept form is given by [tex]$y = mx + b$[/tex], where [tex]$m$[/tex] represents the slope and [tex]$b$[/tex] represents the y-intercept. In the equation [tex]$y = -5x$[/tex], the coefficient of [tex]x \ \ is \ -5[/tex], which represents the slope of the line.

To find the y-intercept, we set [tex]$x$[/tex] equal to [tex]$0$[/tex] and solve for [tex]$y$[/tex]. Substituting [tex]$x = 0$[/tex] into the equation gives us [tex]$y = -5(0) = 0$[/tex].

Hence, the y-intercept of the line [tex]y = -5x \ is \ 0[/tex]. In the graph, the line passes through the origin [tex]$(0,0)$[/tex], indicating that it intersects the y-axis at [tex]$y = 0$[/tex]. This means the line passes through the point [tex]$(0, 0)$[/tex] on the coordinate plane. The y-intercept is the point at which the line crosses the y-axis.

The y-intercept is a crucial point on a line as it helps determine its starting position on the y-axis. In the graph, you can visually observe that the line passes through the origin, confirming the y-intercept of [tex]$0$[/tex].

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The dollar value of the stock that Raquel wants to purchase is $1,833.33.

If the margin requirement is 55%, then the amount of cash wanted is 45% of the total value of the inventory.

Consequently, we will set up the following equation like this:

0.45x = 825

Wherein:

x is the dollar value of the stock Raquel wants to purchase.

To solve for x, we are able to divide both aspects of the equation by way of 0.45:

x = 825 ÷ 0.45

x = $1,833.33

Therefore, the dollar value of the stock that Raquel wants to purchase is $1,833.33.

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[tex]\huge\mathcal{\fcolorbox{Aqua}{azure}{\red{➳Answer}}}[/tex]

Base of the Parallelogram=14m

Height of the Parallelogram=8m

∴ Are of the Parallelogram= Base × height

= (14×8) sq. m

=112 sq.m

Now,

Diameter of the circle (unshaded portion)=8m

Radius of the circle= Diameter/2 =8/2 m= 4m

So, radius=4m

∴ Area of the circle = πr²

=(22/7 × 4 × 4)sq.m

=352/7 sq.m

=50.2857 sq.m

=50.286 sq.m

∴ Are of the Unshaded portion= Are of the Parallelogram - Area of the circle

=(112-50.286) sq.m

= 61.714 sq.m (Ans)

∴The area of the shaded portion is 61.714 sq.m

Hope it helps you

[tex]\red{\rule{200pt}{5pt}}[/tex]

[tex]\bold{Thank ~you~:)}[/tex]

___________________________________

Given:-

Base of the Parallelogram=14m

Height of the Parallelogram=8m

∴ Are of the Parallelogram= Base × height

= (14×8) sq. m

=112 sq.m

Now,

Diameter of the circle (unshaded portion)=8m

Radius of the circle= Diameter/2 =8/2 m= 4m

So, radius=4m

∴ Area of the circle = πr²

=(22/7 × 4 × 4)sq.m

=352/7 sq.m

=50.2857 sq.m

=50.286 sq.m

∴ Are of the Unshaded portion= Are of the Parallelogram - Area of the circle

=(112-50.286) sq.m

= 61.714 sq.m (Ans)

∴The area of the shaded portion is 61.714 sq.m

___________________________________

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Answer: -4x^2-x+15

Step-by-step explanation:

Combine like terms: 2x^2 and -6x^2 combine to -4x^2.

Combine like terms: -4x and 3x combine to -x.

Combine like terms: 8 and 7 combine to 15.

The simplified expression is -4x^2-x+15.