Suppose that water is pouring into a swimming pool in the shape of a right circular cylinder at a constant rate of 5 cubic feet per minute. If the pool has radius 7 feet and height 8 feet, what is the rate of change of the height of the water in the pool when the depth of the water in the pool is 5 feet?

The linear inequality for the shaded region with slope -4 is:

[tex]y < -4x + 4[/tex]

In mathematics, a linear inequality is an inequality involving a linear function in one or more variables. It describes a region in the coordinate plane that satisfies the inequality.

In mathematics, the slope is a measure of the steepness of a line. It describes how much a line rises or falls as we move from left to right along it.

According to the given information,

To write the linear inequality for the graph passing through points (0,4) and (1,0), we need to find the equation of the line first.

The slope of the line passing through these two points is:

[tex]m = (y_{2} - y_{1} ) / (x_{2} - x_{1})[/tex]

= (0 - 4) / (1 - 0)

= -4

Using the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, we can find the equation of the line passing through these two points:

[tex]y = -4x + 4[/tex]

Now, to write the linear inequality for this line, we need to determine which side of the line is shaded. We can use the test point (0,0) to check which side of the line contains the solutions to the inequality.

If we plug in (0,0) into the equation [tex]y = -4x + 4[/tex], we get:

0 = -4(0) + 4

0 = 4

Since 0 is not less than 4, the point (0,0) is not a solution to the inequality. Therefore, we need to shade the side of the line that does not contain the origin (0,0).

The linear inequality for the shaded region is:

[tex]y < -4x + 4[/tex]

So any point below the line [tex]y = -4x + 4[/tex]satisfies this inequality.

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Answer:

[tex]u = \frac{2 \sqrt{6} }{3} [/tex]

[tex]v = \frac{ \sqrt{6} }{3} [/tex]

Step-by-step explanation:

Use trigonometry:

[tex] \tan(60°) = \frac{ \sqrt{2} }{v} [/tex]

Use the property of proportion to find v:

[tex]v = \frac{ \sqrt{2} }{ \tan(60°) } = \frac{ \sqrt{2} }{ \sqrt{3} } = \frac{ \sqrt{2} \times \sqrt{3} }{ \sqrt{3} \times \sqrt{3} } = \frac{ \sqrt{6} }{3} [/tex]

Use the Pythagorean theorem to find u:

[tex] {u}^{2} = {v}^{2} + ( { \sqrt{2} )}^{2} [/tex]

[tex] {u}^{2} = ( { \frac{ \sqrt{6} }{3}) }^{2} + ( { \sqrt{2} )}^{2} = \frac{6}{9} + \frac{2}{1} = \frac{6}{9} + \frac{2 \times 9}{9} = \frac{6}{9} + \frac{18}{9} = \frac{24}{9} = \frac{8}{3} [/tex]

[tex]u > 0[/tex]

[tex]u = \sqrt{ \frac{8}{3} } = \frac{2 \sqrt{6} }{3} [/tex]

The top and lower sums for n =2,4, and 8 and f(x) = 2 +sin(x),0  x   are as follows:

n = 2: Upper Sum = 7.85398; Lower sum ≈ 7.85398

n = 4: Upper sum ≈ 6.43917; Lower sum ≈ 6.43917

n = 8: Upper sum ≈ 6.35258; Lower sum ≈ 6.352

It is necessary to first divide the range [0, ] into n subintervals of identical width x, where x = ( - 0)/n = /n, in order to calculate the upper and lower sums for the equations f(x) = 2 + sin(x), 0 x for n = 2, 4, and 8. The endpoints of these subintervals are:

x0 = 0, x1 = Δx, x2 = 2Δx, ..., xn-1 = (n-1)Δx, xn = π.

Then, for each subinterval [xi-1, xi], we can approximate the area under the curve by the area of a rectangle whose height is either the maximum or minimum value of f(x) on that interval. The sum of these areas' overall subintervals gives us the upper and lower sums.

For n = 2:

[0, π/2]: f(π/2) = 2 + sin(π/2) = 3

[π/2, π]: f(π) = 2 + sin(π) = 2

[0, π/2]: f(0) = 2 + sin(0) = 2

[π/2, π]: f(π/2) = 2 + sin(π/2) = 3

For n = 4:

[0, π/4]: f(π/4) = 2 + sin(π/4) ≈ 2.70711

[π/4, π/2]: f(π/2) = 2 + sin(π/2) = 3

[π/2, 3π/4]: f(3π/4) = 2 + sin(3π/4) ≈ 2.29289

[3π/4, π]: f(π) = 2 + sin(π) = 2

[0, π/4]: f(0) = 2 + sin(0) = 2

[π/4, π/2]: f(π/4) = 2 + sin(π/4) ≈ 2.70711

[π/2, 3π/4]: f(π/2) = 2 + sin(π/2) = 3

[3π/4, π]: f(3π/4) = 2 + sin(3π/4) ≈ 2.29289

For n = 8:

[0, π/8]: f(π/8) = 2 + sin(π/8) ≈ 2.25882

[π/8, π/4]: f(π/4) = 2 + sin(π/4) ≈ 2.70711

[π/4, 3π/8]: f(3π/8) = 2 + sin(3π/8) ≈ 2.96593

[3π/8, π/2]: f(π/2) = 2 + sin(π/2) = 3

[π/2, 5π/8]: f(5π/8) = 2 + sin(5π/8) ≈ 2.96593

[5π/8, 3π/4]: f(3π/4) = 2 + sin(3π/4) ≈ 2.70711

[3π/4, 7π/8]: f(7π/8) = 2 + sin(7π/8) ≈ 2.25882

[7π/8, π]: f(π) = 2 + sin(π) = 2

[0, π/8]: f(0) = 2 + sin(0) = 2

[π/8, π/4]: f(π/8) = 2 + sin(π/8) ≈ 2.25882

[π/4, 3π/8]: f(π/4) = 2 + sin(π/4) ≈ 2.70711

[3π/8, π/2]: f(3π/8) = 2 + sin(3π/8) ≈ 2.96593

[π/2, 5π/8]: f(π/2) = 2 + sin(π/2) = 3

[5π/8, 3π/4]: f(5π/8) = 2 + sin(5π/8) ≈ 2.96593

[3π/4, 7π/8]: f(3π/4) = 2 + sin(3π/4) ≈ 2.70711

[7π/8, π]: f(7π/8) = 2 + sin(7π/8) ≈ 2.25882

The complete question is:-

Unless specified, all approximating rectangles are assumed to have the same width. Evaluate the upper and lower sums for f(x) = 2 + sin(x),0 ≤ x ≤ π with n = 2, 4, and 8.

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Five-number summary: Minimum = 30, Q1 = 35, Median = 50, Q3 = 65, Maximum = 70. Bοx and whisker plοt: Bοx spans frοm 35 tο 65 with median at 50, whiskers extend frοm 30 tο 70, nο οutliers.

Tο find the five-number summary and cοnstruct a bοx and whisker plοt, we need tο first οrganize the given data in ascending οrder:

30, 40, 50, 60, 70

The five-number summary cοnsists οf the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value.

Minimum value: 30

Q1 (first quartile): the median οf the lοwer half οf the data set, which is (30 + 40)/2 = 35

Median (Q2): the middle value οf the data set, which is 50

Q3 (third quartile): the median οf the upper half οf the data set, which is (60 + 70)/2 = 65

Maximum value: 70

Sο, the five-number summary is:

Minimum = 30

Q1 = 35

Median = 50

Q3 = 65

Maximum = 70

To construct a box and whisker plot, we draw a number line that includes the range of the data (from the minimum value to the maximum value), and mark the five-number summary on the number line. Then we draw a box that spans from Q1 to Q3, with a vertical line inside the box at the median (Q2). In addition, we draw "whiskers" from the box to the minimum and maximum values.

The box and whisker plot for the given data is as follows:

       20         40         60         80        100

       |----------|----------|----------|----------|

                   +-----+                    

                   |     |                    

                   |     |                    

                   |     |                    

                   |     |                    

                   +-----+                    

The box spans from 35 to 65, with a vertical line inside the box at 50. The whiskers extend from 30 to 70. There are no outliers in the data, so there are no points beyond the whiskers.

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Answer:

Step-by-step explanation:

Since A and B are independent events, we can use the formula:

P(A ∩ B) = P(A) x P(B)

P(A ∩ B) = 0.10 x 0.60

P(A ∩ B) = 0.06

So the probability of both events A and B occurring is 0.06.

To find P(A U B), we can use the formula:

P(A U B) = P(A) + P(B) - P(A ∩ B)

P(A U B) = 0.10 + 0.60 - 0.06

P(A U B) = 0.64

Therefore, the probability of either A or B occurring (or both) is 0.64.

Answer:

825 miles

Step-by-step explanation:

275 x 3 = 825

Helping in the name of Jesus.

The Griffins ended up paying a total of $575,647.60 for the house. The Griffins paid a total of $331,647.60 in interest over the 30-year period

Interest rate refers to the percentage of the principal amount charged by a lender to a borrower for the use of money over a certain period of time.

(a) The total amount they ended up paying for the house can be calculated by adding the down payment to the total amount paid in monthly mortgage payments over the 30-year period:

Total amount paid = down payment + (monthly payment x number of payments)

Number of payments = 30 years x 12 months/year

                                    = 360

Total amount paid = $48,000 + ($1462.91 x 360)

                               = $48,000 + $527,647.60

                               = $575,647.60

Therefore, the Griffins ended up paying a total of $575,647.60 for the house.

(b) The total amount of interest paid on the mortgage can be calculated by subtracting the amount borrowed (i.e., the purchase price minus the down payment) from the total amount paid over the 30-year period:

Total interest paid = total amount paid - amount borrowed

Amount borrowed = $292,000 - $48,000 = $244,000

Total interest paid = $575,647.60 - $244,000 = $331,647.60

Therefore, the Griffins paid a total of $331,647.60 in interest over the 30-year period.

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Answer:

4567

578९=8877

5790=9766

Answer:

11/15 : 7/18 = 15 / 11 : 18 / 7=  270 / 77

Step-by-step explanation:

g(x) is the translation 5 units right of f(x)= – 7(x–5)²+3.

A function called f(x) accepts an input of "x" and outputs "y". You can write it out as y = f. (x).‘x’ is a variable that represents an input to a function.

To translate a function, we need to replace x with (x-a) in the function f(x) where ‘a’ is the amount of translation.

To translate a function 5 units right, we need to replace x with (x-5) in the function f(x).

So, g(x) = f(x-5) = -7(x-5-5)²+ 3 = -7(x-10)²+ 3.

Therefore, g(x) is the translation 5 units right of f(x)= – 7(x–5)²+3.

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The Triangle Proportionality Theorem, also known as the Side Splitter Theorem, states that if a line is parallel to one side of a triangle, then it divides the other two sides proportionally.

Triangle Proportionality Theorem:

To prove this theorem, we begin by drawing a ΔABC with a line DE parallel to side AB. We then draw lines BD and CE, which intersect the parallel line DE at points F and G, respectively. By the properties of parallel lines, we know that ∠ADE and ∠ABD are congruent, and ∠AED and ∠ADB are congruent. Similarly, ∠CDE and ∠BDC are congruent, and ∠CED and ∠DCB are congruent.

We can then use the properties of similar triangles to show that ΔADE and ΔABC are similar, as are  ΔCDE and ΔACB. This means that the ratios of corresponding side lengths are equal:

BA/DE = CA/CE and CB/DE = AB/BD

We can then substitute CA - BA for CB in the first equation, and BD for AB in the second equation:

BA/DE = (CA - BA)/CE and CB/DE = BD/(CA - BA)

Cross-multiplying both equations, we obtain:

BA * CE = DE * (CA - BA) and CB * DE = BD * (CA - BA)

Adding the two equations, we get:

BA * CE + CB * DE = (DE + CE) * CA

Dividing both sides by CB * DE, we obtain:

BA/CB = (DE + CE)/CE * CA/DE = DE/CD

Thus, we have proven the Triangle Proportionality Theorem.

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the frequency of the sinusoidal graph is 1/π.

The new average handling time for Agent D would be 90% of their original handling time.

A percentage is a way of expressing a number as a fraction of 100. It is typically represented using the percent sign (%) and is often used to describe the amount or proportion of something in relation to a whole. Percentages are commonly used in a variety of fields, such as finance, mathematics, and statistics.

Let's say the original average handling time for Agent D was "x" units (e.g. seconds, minutes, etc.). If Agent D was able to reduce their average handling time by 10%, their new average handling time would be:

New average handling time = x - 0.1x

Simplifying the expression on the right-hand side:

New average handling time = 0.9x

Therefore, the new average handling time for Agent D would be 90% of their original handling time.

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Answer:

Step-by-step explanation:

Starting with 72x5/12:

72x5/12 = (72/12) x 5 (simplifying the fraction)

= 6 x 5

= 30

Now, we have:

30 = ?x5x1/12

Multiplying both sides by 12, we get:

30 x 12 = ? x 5 x 1

360 = ? x 5

Dividing both sides by 5, we get:

72 = ?

Therefore, the missing value is 72.

After answering the presented question, we can conclude that the probability of 30 or more seconds between vehicle arrivals is approximately 0.082.

Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many fields, including statistics, economics, science, and engineering.

exponential probability distribution

[tex]P(X ≤ 12) = ∫[0,12] f(x) dx = ∫[0,12] (1/12) * e^(-x/12) d\\P(X ≤ 12) = [-e^(-x/12)] [0,12] = -e^(-1) + 1 ≈ 0.632\\P(X ≤ 6) = ∫[0,6] f(x) dx = ∫[0,6] (1/12) * e^(-x/12) dx\\P(X ≤ 6) = [-e^(-x/12)] [0,6] = -e^(-1/2) + 1 ≈ 0.393\\[/tex]

Therefore, the probability that the arrival time between vehicles is 6 seconds or less is approximately 0.393.

P(X ≥ 30) = 1 - P(X < 30) = 1 - P(X ≤ 30) = 1 - ∫[0,30] f(x) dx

[tex]= 1 - ∫[0,30] (1/12) * e^(-x/12) dx[/tex]

[tex]P(X ≥ 30) = 1 - [-e^(-x/12)] [0,30] = e^(-2.5) ≈ 0.082[/tex]

Therefore, the probability of 30 or more seconds between vehicle arrivals is approximately 0.082.

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a. The revenue function R(x) = 75x - 3x² graph plotted below.

b. Value of x is 12.5 and maximum revenue produces $468.75

c. Wholesale price/chip that produces the maximum revenue is $37.50

A company's or business's revenue is the total amount of money it receives from sales of its products or services over a given time period.

a) To sketch a graph of the revenue function R(x), we first need to calculate R(x) using the given formula:

⇒ R(x) = x P(x)

⇒ R(x) = x(75 - 3x)

⇒ R(x) = 75x - 3x²

Now we can plot this function on a rectangular coordinate system. Below is an sketch of the graph.

b) Taking the derivative of R(x) and setting it equal to 0:

⇒ R'(x) = 75 - 6x = 0

⇒ 6x = 75

⇒  x = 12.5

So, x = 12.5 is the value of x that will produce the maximum revenue. To find the maximum revenue, we can substitute x = 12.5 into the revenue function R(x) = x (75 - 3x)

⇒ R(12.5) = 12.5 (75 - 3 (12.5)) = 468.75

⇒ R(12.5) = $ 468.75

Therefore, the maximum revenue is $468.75

c) To find the wholesale price per chip that produces the maximum revenue, we can substitute x = 12.5 into the price function, P(x) = 75 - 3x​

⇒ P(12.5) = 75 - 3(12.5)

⇒ P(12.5) =  $37.50 per chip

Therefore, the wholesale price per chip that produces the maximum revenue is $37.50.

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Answer: C (23)

Step-by-step explanation:

Since the triangle is isosceles, BA = BC

x + 17 = 2x -6

x = 23

Answer:

C(23)

Step-by-step explanation:

Since line AB = line BC

x+17=2x-6, by collecting like terms x=23

Answer:

False

Step-by-step explanation:

Purple: 13÷8= 1.625

Teal: 15÷6= 2.5

The ratios don't have the same scale factor. Therefore they are not equivalent fractions.

sin 45° = opposite/hypotenuse = x/√2x = √2/2. The answer is B. √2/2.

What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It focuses on the study of trigonometric functions, which are functions that relate the angles of a triangle to the ratios of the lengths of its sides.

The triangle shown in the diagram is a right triangle with one angle of 45 degrees, which means that the other two angles must measure 45 degrees each as well.

The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we can label the sides of the triangle as follows:

The side opposite the 45-degree angle is x.

The side adjacent to the 45-degree angle (and opposite the other 45-degree angle) is also x.

The hypotenuse is the longest side of the triangle and is labeled as √2x.

Using the definition of sine, we have:

sin 45° = opposite/hypotenuse = x/√2x = √2/2

Therefore, the answer is B. √2/2.

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On solving the provided question we can say that As a result, the triangle is resolved.

The study of the relationship between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their potential applications in computations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle properties, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric shapes.

We now have all of the information we require to solve the triangle. We now have:

a = 55.815 sin(A) (A

b = 55.815 sin(B) (B)

c = 32

A + B + C = 180°

B = 153° - A

To find the values of A and B, we can use a calculator. We get:

A ≈ 83.814°

B ≈ 42.186°

32 / sin(27°) = a / sin(A)

a ≈ 54.482

AB ≈ 54.482

BC ≈ 39.343

AC ≈ 22.414

A ≈ 83.814°

B ≈ 42.186°

C ≈ 27°

As a result, the triangle is resolved.

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After solving by linear programming, the business needs create 100 Type I pencils and 80 Type II pencils to increase revenue.

LINEAR PROGRAMMING: WHAT IS IT?

A mathematical method called linear programming is used to maximise a linear objective function under the restrictions of linear equality and inequality. In a mathematical model whose requirements are expressed by linear connections, it is used to identify the best result.

With linear programming, this issue can be resolved.

Please define x as the quantity of Type I pencils produced and y as the quantity of Type II pencils produced.

Profit = 3x + 5y is the formula for the goal function.

2x + 5y 660 are the restrictions (sanding constraint)

(Polishing constraint): x ≥0 y≥ 0; 6x + 3y 730

Under these limitations, we wish to maximize the profit function.

The largest profit comes when x = 100 and y = 80,

with a profit of $740, according to software that can solve linear programming issues or a graphing calculator.

Thus, the business needs create 100 Type I pencils.80pencil for type II

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The experimental probability that the next student will register for German is 9/79.

What is probability?

To find the experimental probability that the next student will register for German, we need to divide the number of students who have registered for German by the total number of students who have registered so far:

P(German) = number of students who have registered for German / total number of students who have registered

P(German) = 108 / (108 + 360 + 21 + 459) [Adding all the students who registered for each language]

P(German) = 108 / 948

P(German) = 9/79

Therefore, the experimental probability that the next student will register for German is 9/79.

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Answer: (7.2, -3.2)

Step-by-step explanation:

         First, we will graph these equations. See attached. One has a y-intercept of -5 and then moves four units right for every unit up (we get this from the slope of 1/4). The other has a y-intercept of 4, and moves right one unit for every unit down (we get this from the slope of -1).

         The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:

                         (7.2, -3.2)     x = 7.2, y = -3.2

Part A: The system of equations represented by the graph: y = 3x - 5 and 2x + 4y = 8.

Part B: The solution of the system of equations : (2, 1).

Determining the significance of the variables employed in a system of equations entails solving the set of equations.

A specific system of equations may have a variety of solutions,

Let's examine three approaches to solving a set of equations, presuming that they are linear equations with two variables.

From the graph shown.

The blue line shows the equation: 2x + 4y = 8

At x =0, y= 2

At y =0, x = 4

Red line shows the equation: y = 3x - 5

At x = 0, y = -5.

Part B: solution to the system of equations.

From the graph, where two lines intersect is the solution of the system of equations.

That is point (2,1).

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So the area of the living room on the scale drawing is 334128.48 square centimeters.

Area is a measure of the size of a two-dimensional surface or region, typically expressed in square units. It is the amount of space inside a flat, enclosed shape or surface, and is calculated by multiplying the length and width of the shape or surface. For example, the area of a rectangle can be calculated by multiplying its length by its width, while the area of a circle can be calculated by multiplying pi (3.14) by the square of its radius. Area is a fundamental concept in mathematics and is used in a wide range of fields, from geometry and physics to engineering and architecture.

Here,

First, we need to determine the actual dimensions of the living room. Since each 4 cm on the scale drawing is equal to 8 feet, we can set up a proportion:

4 cm : 8 feet = 12 cm : x

Solving for x, we get:

x = (12 cm x 8 feet) / 4 cm

= 24 feet

So the actual length and width of the living room are 24 feet and 24 feet, respectively.

The area of the living room is then:

Area = length x width

= 24 feet x 24 feet

= 576 square feet

Now, we need to determine the area of the living room on the scale drawing. Since the length and width of the scale drawing are both 12 cm, the area is:

Area = length x width

= 12 cm x 12 cm

= 144 square cm

Finally, we can determine the scale factor for the area by dividing the actual area by the scale area:

Scale factor = actual area / scale area

= 576 square feet / 144 square cm

Since we need the area in square centimeters, we can convert square feet to square centimeters by multiplying by 929.03:

Scale factor = (576 square feet / 144 square cm) x (929.03 square cm/square feet)

= 2324.12

Therefore, the area of the living room on the scale drawing is:

Area = scale area x scale factor

= 144 square cm x 2324.12

= 334128.48 square cm

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The range, on the other hand, is affected by extreme values and may not be a good representation of the spread of the data in these cases.

What is Histogram ?

A histogram is a graphical representation of the distribution of a dataset. It is a way to display the frequency of occurrence of different values or ranges of values in a dataset.

The correct answer is IQR, because it is more robust to outliers and is not affected by extreme values like Range.

Although the question provides information about the shape of the histograms, it does not indicate whether the distributions are symmetric or skewed. Therefore, the choice of IQR over Range is not based on the shape of the data but on the fact that IQR is a more appropriate measure of variability when dealing with skewed data or data with outliers.

In general, the IQR is a better measure of variability than the range when the data is skewed or contains outliers, as it only considers the middle 50% of the data and is not affected by extreme values.

Therefore, The range, on the other hand, is affected by extreme values and may not be a good representation of the spread of the data in these cases.

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Answer:

Step-by-step explanation:

23 people ordered dessert.

8 of these ordered only a dessert.

P(not starter | ordered dessert) [tex]=\frac{8}{23}[/tex]

Answer:

The answer to your question is $2.55

Step-by-step explanation:

85% × $3 = $2.55

With original price $3 and 85% off,

Final price: $0.45

Saved amount: $2.55

I hope this helps and have a wonderful day!

Answer:0.45

Step-by-step explanation:

Purchase Price:

$3

Discount:

(3 x 85)/100 = $2.55

Final Price:

3 - 2.55 = $0.45

Answer: The perimeter of a rectangle is found by adding up all four sides. For this rectangle with a base of 9 ft and a height of 10 ft, the two base sides have a length of 9 ft each, and the two height sides have a length of 10 ft each. Therefore, the perimeter is:

P = 2(9 ft) + 2(10 ft) = 18 ft + 20 ft = 38 ft

So the perimeter of the rectangle is 38 feet.

Step-by-step explanation:

Answer: Jake solved it incorrectly.

Step-by-step explanation: His equation was supposed to be (7/2*35/2) *18