Answer options are

A. 6
B. 8
C. 4
D. 0

Answer:

For both problems, the correct equation of the line will be presented in three forms.

14)

y - 4 = -5(x + 1)---point-slope form

y - 4 = -5x - 5

y = -5x + 1---slope-intercept form

5x + y = 1---standard form

15)

y + 3x = 13

y = -3x + 13

y - 10 = (1/3)(x - 6)---point-slope form

y - 10 = (1/3)x - 2

y = (1/3)x + 8---slope-intercept form

3y = x + 24

-x + 3y = 24---standard form

x - 3y = -24

Answer & Step By Step Explanation:

a) The function D(x) = 72x represents the distance traveled by the driver after x hours. To find out how far the driver is from the rest stop after 3 hours, we simply substitute x = 3 into the function:

[tex]D(3) = 72 * 3 = 216 miles[/tex]

So, the driver is 216 miles away from the rest stop after 3 hours.

b) The slope of the graph of D(x) = 72x is 72. In the context of this problem, the slope represents the rate of change of the distance with respect to time. This is essentially the speed of the driver. So, the driver is traveling at a constant speed of 72 miles per hour.

Considering the triangles, the value of BE is 27

This is a term used in geometry to mean that the respective sides of the triangles are proportional and the corresponding angles of the triangles are congruent

In the figure, BE is solved using the following ratio

FD / FA = DC / AE

1 / 2 = 27 / AE

AE = 27 * 2

AE  = 54

BE = 1/2 AE

BE = 27

hence BE = 27

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Answer:

84

Step-by-step explanation:

A= [tex]\frac{1}{2}[/tex]b*h

A= [tex]\frac{1}{2}[/tex]14*12

A=84

Answer:

The lateral surface area of the cone is about 147.2 ft².

The surface area of the sides and the bottom of the cylinder is about 867.1 ft².

The total surface area of the figure is about 1014.3 ft².

Step-by-step explanation:

The formula for the lateral surface area of a cone is:

[tex]\boxed{\begin{minipage}{7 cm}\underline{Lateral Surface Area of a Cone}\\\\$L.S.A.=\pi r \sqrt{r^2 + h^2}$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius of the circular base. \\ \phantom{ww}$\bullet$ $h$ is the perpendicular height of the cone.\\\end{minipage}}[/tex]

From inspection of the given diagram (attached), we can see that the diameter of the base of the cone is 12 ft.

As the radius of a circle is half its diameter, the radius of the base of the cone is r = 6.

The height of the cone is 5 ft, so h = 5.

Substitute the values of r and h into the formula to calculate the lateral surface area of the cone:

[tex]\begin{aligned}\textsf{Lateral Surface Area of the Cone}&=\pi (6) \sqrt{6^2 + 5^2}\\&=6\pi \sqrt{36 +25}\\&=6\pi \sqrt{61}\\&=6\sqrt{61}\pi \\&=147.219738...\\&=147.2\; \sf ft^2\;(nearest\;tenth)\end{aligned}[/tex]

The surface area of the cylinder part of the figure is made up of the lateral surface area of the cylinder and the area of one circular base.

Therefore the formula to use to calculate the area of the sides and bottom of the cylinder is:

[tex]\boxed{L.S.A.=2\pi r h+\pi r^2}[/tex]

where r is the radius and h is the height of the cylinder.

From inspection of the given diagram, we can see that the radius of the cylinder is r = 6 and the height is h = 20. Substitute these into the formula to find the area of the sides and bottom of the cylinder:

[tex]\begin{aligned}\textsf{S.A. side and bottom of cylinder}&=2 \pi (6)(20)+\pi (6)^2\\&=2 \pi (120)+\pi (36)\\&=240 \pi +36\pi\\&=276 \pi \\&=867.079572...\\&=867.1\; \sf ft^2\;(nearest\;tenth)\end{aligned}[/tex]

The total surface area of the figure is the sum of the lateral surface area of the cone and the sides and bottom of the cylinder. Therefore:

[tex]\begin{aligned}\textsf{Total Surface Area}&=147.219738...+867.079572...\\&=1014.29931...\\&=1014.3\; \sf ft^2\;(nearest\;tenth)\end{aligned}[/tex]

In conclusion:

Answer:

h=1.8 cm

Step-by-step explanation:

Solution Given:

Volume of triangular prism= Area of base * length

Here

Volume of traingular Prism= 63 cm³

base =7 cm

length =10 cm

Area of base = Area of triangle=½*base*height=½*7*h

Now

substituting value in the formula:

63= ½*7*h*10

63=35*h

h=63/35

h=1.8 cm

Therefore, height of the triangular cross section is 1.8 cm.

The degree of the given polynomial is 4.The given polynomial is 2x²y - 8xy³ + 5xy. Let us count the number of terms in this polynomial. We have 3 terms.Let us now calculate the degree of each of these terms.

The degree of the first term is (2 + 1) = 3, as there are two variables x and y, and both are raised to the power 2 and 1 respectively.The degree of the second term is (1 + 3) = 4, as there are two variables x and y, and x is raised to the power 1 and y to the power 3.

The degree of the third term is (1 + 1) = 2, as there are two variables x and y, and both are raised to the power 1 each. So the degree of this polynomial will be the highest degree among these 3 terms.

The highest degree is 4. Therefore, the degree of the given polynomial is 4.

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Answer:

c = 50

Step-by-step explanation:

2c - 3 = 97

2c = 100

c = 50

The company should produce and sell 60 items to maximize its profit. The profit at this level of production will be $2,350.

For a certain company, the cost function for producing x items is C(x) = 40 x + 150 and the revenue function for selling x items is R(x) = -0.5(x - 100)2 + 5,000. The maximum capacity of the company is 160 items.To find the optimal number of items that the company should produce and sell, we need to maximize the revenue function R(x) while also taking into account the cost function C(x).

The profit function P(x) can be expressed as:P(x) = R(x) - C(x).

Substituting the given equations, we get:P(x) = (-0.5(x - 100)2 + 5,000) - (40x + 150)P(x) = -0.5(x - 100)2 - 40x + 4,850.

To maximize the profit, we need to find the value of x that gives the maximum value for P(x). The maximum capacity of the company is 160 items, so we only need to consider the values of x from 0 to 160.

To find the optimal value of x, we can take the derivative of P(x) and set it equal to zero: d P(x)/dx = -x + 100 - 40 = 0.

Solving for x, we get: x = 60. This means that the optimal number of items that the company should produce and sell is 60. At this level of production, the profit will be:P(60) = -0.5(60 - 100)2 - 40(60) + 4,850 P(60) = $2,350.

Therefore, the company should produce and sell 60 items to maximize its profit. The profit at this level of production will be $2,350.

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a. Domain -   [0, 20]

b. Range -   [8, ∞)

c. X-intercepts -   undefined.

d. Y-intercept -   8

e. Missing function value f(9) = 11

a. The function's domain is the range of x-values over which the graph is defined.  In this case, the domain is from 0 to 20, inclusive -   [0, 20].

b. The function's range is the set of all possible y-values that the function takes. Since the graph starts from unit 8 on the y-axis, the range would be from 8 to positive infinity -   [8, ∞).

c. since the   graph does not touch the x- axis at any point, we can state that the there is no x-intercept orthat the x -intercept is undefined.

d. The y-intercept is the point where the   graph intersects the y-axis. In this case, the y-intercept is at unit 8 on the y-axis.

e.  Based on the graph the point (9, 11) corresponds to the value of y when x is 9, then f(9) would be equal to 11.

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The estimated total cost of the project is $7,663.20.

We need to add up the costs of soil, sand, sod, and tree/shrubbery installation, including the contingency.

Soil: 8 cubic yards x $80/cubic yard = $640

Sand: 3 cubic yards x $100/cubic yard = $300

Sod: 3600 square feet / 30 square feet/roll x $100/roll = $1200

Tree and shrubbery installation: $2500

Contingency: 3% x $4640 = $139.20

Total cost: $640 + $300 + $1200 + $2500 + $139.20 = $7,663.20

Therefore, The estimated total cost of the project is $7,663.20

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Answer: A= -1, B= -3,C=-4.D=3 i think

The correct option is C, ON is the normal to the surface.

Remember that when we have an incident ray on a surface, a reflected ray will be emmited such that the angle with respect to the normal is the same one for the incident ray.

For any plane surface, this normal is perpendicular to the surface.

Here, we can see that ON is normal to the sruface of reflection, thus, the correct option is C, segment ON is the normal.

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The lengths of the other two sides of the right triangle are 24m and 25m, respectively.

Let's assume the consecutive integers representing the lengths of the other two sides of the right triangle are x and x + 1, where x is the smaller integer. We are given that the shortest side measures 7m. Now, we can use the Pythagorean theorem to solve for the lengths of the other two sides.

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

Using this theorem, we have the equation:

[tex]7^2 + x^2 = (x + 1)^2[/tex]

Expanding and simplifying this equation, we get:

[tex]49 + x^2 = x^2 + 2x + 1[/tex]

Now, we can cancel out [tex]x^2[/tex] from both sides of the equation:

49 = 2x + 1

Next, we can isolate 2x:

2x = 49 - 1

2x = 48

Dividing both sides by 2, we find:

x = 24

Therefore, the smaller integer representing the length of one side is 24, and the consecutive integer representing the length of the other side is 24 + 1 = 25.

Hence, the lengths of the other two sides of the right triangle are 24m and 25m, respectively.

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Answer:

The answer is approximately 11 stories

Step-by-step explanation:

6 stories------->22 meters

x stories---------->40 meters

[tex]x = \frac{6 \times 40}{22} [/tex]

x≈11 stories

Answer:

Step-by-step explanation:

Let's assume that Leon has 'x' number of coloured pencils. According to the question, Box A contains 3/8 of all his coloured pencils.So, Box B contains (1-3/8) = 5/8 of all his coloured pencils.  Therefore, number of pencils in Box A = (3/8) * x  Also, number of pencils in Box B = (5/8) * x  According to the question, Leon removes 2/5 of the pencils from Box B to colour a picture. So, pencils used from Box B = (2/5) * (5/8) * x = (1/4) * x  And the total number of pencils used for the picture = 15.  So, pencils used from Box A = 15 - (1/4) * x  Now, the total pencils used in the picture = pencils used from Box A + pencils used from Box B  Therefore, 15 = 15 - (1/4) * x + (1/4) * x  Thus, x = 60  Therefore, number of pencils in Box A = (3/8) * 60 = 22.5.  Hence, there are 22.5 pencils in Box A.

The simplified boolean expression for F5 is F5 = P OR (NOT R). In this expression, P represents one boolean variable, and (NOT R) represents the negation of another boolean variable R. The OR operator combines the two variables, resulting in the final boolean expression F5.

To simplify the boolean expression F5 = (P AND Q) OR (R AND NOT P), we can apply Boolean algebra rules to reduce it to its simplest form.

Step 1: Apply De Morgan's laws

NOT (R AND NOT P) can be simplified as (NOT R) OR P.

Step 2: Distributive property

F5 = (P AND Q) OR ((NOT R) OR P)

Step 3: Apply associative property

F5 = (P AND Q) OR (P OR (NOT R))

Step 4: Apply absorption law

F5 = P OR (P OR (NOT R))

Step 5: Apply idempotent law

F5 = P OR (NOT R)

By simplifying the expression, we have reduced it to its simplest form while preserving its logical meaning. The simplified expression can be used to analyze and evaluate logical circuits or systems where the variables P and R are used.

Remember, boolean algebra allows us to manipulate and simplify complex logical expressions by applying various rules and properties. These simplifications help in understanding and analyzing logical operations in digital systems and circuits.

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The question probable may be:

Consider the boolean variables P, Q, and R. Determine the boolean expression for the variable F5 using the given variables:

F5 = (P AND Q) OR (R AND NOT P)

Using the boolean variables P, Q, and R, can you simplify the expression F5 to its simplest form by applying Boolean algebra rules?

The graph that represents the piecewise function, h(x) is the second option

Please find attached the graph that represents h(x) created with MS Excel

A piecewise function comprises of two or more functions, which are defined based on specified intervals of the input variable.

The possible piecewise function that is represented by h(x), obtained from a similar question on the site can be presented as follows;

[tex]h(x) \begin{cases} \frac{1}{4}\cdot x-4, & \text{ } x\leq0 \\\frac{1}{3}\cdot x - 3, & \text{ } 0 < x \leq 3 \\\frac{1}{2}\cdot x-2, & \text{ } x\geq4\end{cases}[/tex]

Therefore, at x ≤ 0, the y-intercept is -4, for h(x) = (1/4)·x - 4, which corresponds to the second graph.

The range for h(x) at 0 < x ≤ 3 is; h(0) = (1/3)×0 - 3 < h(x) ≤ (1/3)×3 - 3, which can be expressed as; (-3, -2], also corresponding to the second graph

The value of x-intercept of h(x) in the interval x ≥ 4 can be found from the equation; h(x) = (1/2)·x - 2 = 0

(1/2)·x = 2

x = 2 × 2 = 4

The x-intercept, (4, 0), corresponds to the second graph, therefore, the graph that represents h(x) is the second graph

Please find attached the graph that represents the piecewise function, h(x), created with MS Excel

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The method that you find important to enhance the strong number sense is  the use of manipulatives and hands-on activities.

Example:

Illustrating numbers and executing mathematical operations through block representations can assist students in visualizing addition, subtraction, and multiplication.

Using manipulatives and interactive activities, numerical concepts can be grasped more tangibly, as learners physically engage with these objects, thereby developing a concrete understanding of mathematics.

Students can enhance their comprehension of measurement and units by employing measurement instruments such as rulers and measuring tapes.

Through participation in these tasks, students cultivate a profound sense of intuition in regard to numbers, operations, and their interconnections, thereby establishing a firm groundwork.

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There are 52,406,250 ways to award the first, second, and third prizes in a contest with 375 contestants, assuming there are no ties.

To solve this problem, we will use permutations.

A permutation is a way of arranging objects or elements in a specific order.

In this case, we want to find the number of permutations of 375 objects taken 3 at a time, because we are selecting three prize winners.

The formula for permutations is: P(n, r) = n!/(n-r)!where n is the total number of objects and r is the number of objects we are selecting.

So in our case, we want to find the number of ways to select 3 prize winners out of 375 contestants:

P(375, 3) = 375!/(375-3)!P(375, 3) = 375!/372!P(375, 3) = 375*374*373P(375, 3) = 52,406,250

Therefore, there are 52,406,250 ways to award the first, second, and third prizes in a contest with 375 contestants, assuming there are no ties.

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Answer:

(0,a)

Step-by-step explanation:

The y-intercept of a function is the point where the graph of the function intersects the y-axis.

To find the y-intercept of the function F(x) = (a)(b^x), we can substitute x = 0 into the equation and solve for F(0).

When x = 0, we have:

=> F(0) = (a)(b^0)

=>  F(0) = (a)(1)

=> F(0) = a

Therefore, the y-intercept of the function F(x) = a * b^x is the point (0, a).

The calculated value of the integral [tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx[/tex] is 0.7358

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

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx[/tex]

The above expression can be integrated using integration by parts method

When integrated, we have

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx = -\left(x+1\right)\mathrm{e}^{-x}[/tex]

Recall that the x values are from 1 to ∝

This means that

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx = -\left(\infty +1\right)\mathrm{e}^{-\infty} + \left(1+1\right)\mathrm{e}^{-1}[/tex]

So, we have

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx = 0 + \frac{2}{e}[/tex]

Solving further, we get

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx = 0 + 0.7358[/tex]

Evaluate

[tex]\int\limits^{\infty}_1 {(xe^{-x}) \, dx = 0.7358[/tex]

Hence, the value of the integral is 0.7358

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The value of k(3) in the function given is 37

Given the function :

To find k(3) ; substitute t = 3 into the equation

k(3) = 13(3) - 2

k(3) = 39 - 2

k(3) = 37

Therefore, the value of k(3) would be 37.

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The correct answer is Option B. 50 chairs can John make in 5 hours from the graph.

John is a furniture maker and the graph shows the number of chairs John makes and the time it takes to make those chairs.

Based on the graph, the number of chairs John can make in 5 hours can be found by following the given steps:

Step 1: Look for the point on the graph where the line intersects the vertical line marking 5 hours.

Step 2: From the point on the graph found in step 1, read the horizontal line and find the point at which it intersects the y-axis.

This point will give us the number of chairs John can make in 5 hours.

According to the graph, John makes 50 chairs in 5 hours.

Therefore, the correct answer is option (B) 50.

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Answer:

x = 26

Step-by-step explanation:

Since s║t and r is a transversal, consecutive interior angles add to 180

⇒ 2(x + 15) + (3x + 20) = 180

⇒ 2x + 30 + 3x + 20 = 180

⇒ 5x + 50 = 180

⇒ 5x = 180 - 50

⇒ 5x = 130

⇒ x = 130/5

⇒ x = 26

Answer:

x=26

Step-by-step explanation:

Given:

s || t

2(x+15) + (3x+20) = 180°

Since the sum of co interior angle is 180°

Opening bracket

2x + 30 + 3x + 20 =180

Solving like terms

5x+50= 180

subtracting both side by 50

5x=180-50

5x = 130

dividing both side by 5.

[tex]\tt x=\frac{130}{5}[/tex]

x = 26

Therefore value of x is 26.

Answer:

Step-by-step explanation:

x+9

Let x, be 4

4+9=13

given condition,

x+9<4x

4+9<4(4)

13<16

The answer is:

Work/explanation:

Our inequality is:

[tex]\sf{x+9 < 4x}[/tex]

Flip it

[tex]\sf{4x > x+9}[/tex]

Solve

[tex]\sf{4x-x > 9}[/tex]

Combine like terms

[tex]\sf{3x > 9}[/tex]

Divide each side by 3

[tex]\sf{x > 3}[/tex]

Drake is correct, and Selena is incorrect in this scenario.Option D.

Selena's Work:

Selena substituted the values for the variables r = -1 and s = -2 directly into the expression before simplifying. She then simplified the expression and evaluated it.

However, this approach is incorrect because the order of operations requires simplification before substitution. Selena should have simplified the expression first and then substituted the values of r and s.

Drake's Work:

Drake followed the correct order of operations. He simplified the expression first and then substituted the values of r = -1 and s = -2. By simplifying the expression, Drake obtained (-1/-2), which simplifies to 1/2. After substituting the values, Drake evaluated the expression correctly and arrived at the final result of 1/2.

Since Drake followed the correct order of operations and obtained the correct result, he is the one who is correct. Selena's mistake was in not simplifying the expression before substituting the values for r and s.

In mathematical operations, it is important to follow the order of operations (PEMDAS/BODMAS) to ensure accurate results. The correct order is to simplify the expression first, and then substitute the given values for the variables. So Option D is correct.

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The value that is the input of the function is 4.

A function relates input and output. In other words,  a function defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

The independent variable is the x values or the input value while the dependent variable is the y values of the output values.

Hence, the values of the input(x) of the function are 4, 6, 8, 10. 12 etc.

Therefore, base on the option, the input of the function is 4.

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The value of is 2/11

To determine the value, we have;

Index forms are mathematical forms used to represent numbers or values that are too large or too small in more convenient forms.

From the information given, we have the expression written as;

2⁹ˣ+2 = 16⁵ˣ -2

Put all the terms in one base form, we have;

2⁹ˣ + 2¹ = 2²⁰ˣ - 2¹

equate the exponents, we have;

9x + 1 = 20x - 1

collect the like terms, we have;

9x - 20x = -2

-11x = -2

Make 'x' the subject of formula, we have;

x = 2/11

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The solution to this integral [tex]\int\limits^{\infty} _{-\infty} e^{-x^{2} } \, dx =\sqrt{\pi}[/tex] is equal to √π.

In Mathematics, the Gaussian integral is sometimes referred to as the probability integral, and it highly related to the erf function, which represents the integral of the one-dimensional Gaussian function over the interval [-∞, ∞].

Generally speaking, the given integral can be computed by using the advanced techniques of combining two one-dimensional Gaussians and converting into polar coordinates;

[tex]I^2 =\int\int e^{(-(x^2 +y^2))}dxdy\\\\I^2 =\int\int e^{-r^2}rdr d\theta[/tex]

In this context, we would use these integration limits [0, ∞) with respect to "r" and [0, 2π] with respect to "θ" as follows:

[tex]I^2 =\int\limits^{2 \pi}_{0} \int \limits^{\infty}_{0} e^{-r^2}rdr d\theta\\\\I^2 =\int\limits^{2 \pi}_{0} \int \limits^{\infty}_{0}\frac{1}{2} e^{-u}du d\theta\\\\I^2 =\int\limits^{2 \pi}_{0} [\frac{1}{2} e^{-u}]\limits^{\infty}_{0}d\theta\\\\I^2 =\int\limits^{2 \pi}_{0} (\frac{1}{2} )d\theta\\\\I^2=\frac{1}{2}[\theta]\limits^{2 \pi}_{0}\\\\I^2=\frac{1}{2}(2 \pi)-\frac{1}{2}(0)\\\\I^2=\frac{2 \pi}{2}[/tex]

I² = π

I = √π

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