Find the surface area AND volume of the following polyhedra. (SHOW ALL
WORK)
8

(a) The diver hit the water after 2.5 seconds

(b) The diver hit the water with a velocity of 80 ft/s

Since the cliff diver plunges from a height of 100 ft above the water surface and the distance the diver falls in seconds is given by the function d(t) = 16t². We can say d(t) = 100ft. So we have:

d(t) = 16t² = 100

t² = 100/16

t² = √(100/16)

t = 10/4

t = 2.5 seconds

The derivative of the distance will give us the velocity. That is:

v(t) = d'(t) = 32t

Substitute t =2.5  into v(t):

v(2.5) = 32(2.5)

v(2.5) = 80 ft/s

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Answer:

See below for answers and explanations

Step-by-step explanation:

Part A

[tex]x^2+4x+4\\\rightarrow x^2+2x+2x+4\\\rightarrow x(x+2)+2(x+2)\\\rightarrow (x+2)(x+2)[/tex]

Part B

[tex]x^2-12x+27\\\rightarrow x^2-9x-3x+27\\\rightarrow x(x-9)-3(x-9)\\\rightarrow (x-3)(x-9)[/tex]

Part C

[tex]x^2-x-42\\\rightarrow x^2+6x-7x-42\\\rightarrow x(x+6)-7(x+6)\\\rightarrow (x-7)(x+6)[/tex]

Part D

[tex]x^2+8x-9\\\rightarrow x^2+9x-x-9\\\rightarrow x(x+9)-1(x+9)\\\rightarrow (x-1)(x+9)[/tex]

Answer: The rate of change for Miquel's position in the race can be calculated by his speed, which is 18 kilometers per hour. So, his rate of change is 18 kilometers per hour. This means that for every hour that passes, Miquel will have covered 18 kilometers in the race.

Step-by-step explanation:

The total weight in the bag is given as follows:

8.2 pounds.

The total weight is obtained applying the proportions in the context of the problem.

From the table, the weight of each equipment is given as follows:

Hence the total weight of a bag with 4 balls, 2 gloves, and 2 bats is given as follows:

4 x 0.35 + 2 x 0.77 + 2 x 2.63 = 8.2 pounds.

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Answer: c[tex]\leq[/tex]-17/4

Step-by-step explanation:

-8c+13[tex]\geq[/tex]47

-8c[tex]\geq[/tex]47-13

-8c[tex]\geq[/tex]34

c[tex]\leq[/tex] -17/4 or -4 1/4

x int = 15

y int = -10

y intercept can be found bt setting X to 0

x intercept can be found by setting Y to 0

2(0) - 3y = 30 --> -3y = 30 --> y = -10

2x - 3(0) = 30 --> 2x = 30 --> x = 15

alternatively, y int is "b" in eq. y = mx + b

-3y = 30 -2×

y = -10 + (2/3)x

y = (2/3)x - 10

b = -10

The probability of all three of the selected homes having a security system is 0.3277, the probability of none of the three selected homes having a security system is 0.0414, and the probability of at least one of the selected homes having a security system is 0.9709. The events are dependent.

The probability of all three of the selected homes having a security system is 0.3277, which can be calculated using the formula P(A and B and C) = P(A) x P(B) x P(C). In this case, P(A) is the probability of the first home having a security system, which is 8/11 (since 8 out of 11 homes have a security system). P(B) is the probability of the second home having a security system, which is 7/10 (since 7 out of 10 remaining homes have a security system). P(C) is the probability of the third home having a security system, which is 6/9 (since 6 out of 9 remaining homes have a security system). Therefore, P(A and B and C) = (8/11) x (7/10) x (6/9) = 0.3277.

The probability of none of the three selected homes having a security system is 0.0414, which can be calculated using the formula P(A and B and C) = P(A') x P(B') x P(C'). In this case, P(A') is the probability of the first home not having a security system, which is 3/11 (since 3 out of 11 homes do not have a security system). P(B') is the probability of the second home not having a security system, which is 3/10 (since 3 out of 10 remaining homes do not have a security system). P(C') is the probability of the third home not having a security system, which is 3/9 (since 3 out of 9 remaining homes do not have a security system). Therefore, P(A and B and C) = [tex](3/11) x (3/10) x (3/9) = 0.0414[/tex].

The probability of at least one of the selected homes having a security system is 0.9709, which can be calculated using the formula P(A or B or C) = 1 - P(A' and B' and C'). In this case, P(A' and B' and C') is the probability of none of the three selected homes having a security system, which is 0.0414 (as calculated above). Therefore, P(A or B or C) = 1 - 0.0414 = 0.9709.

The events are dependent because each selection is affected by the previous selections. For example, the probability of the second home having a security system is affected by whether or not the first home had a security system. If the first home had a security system, then there are 7 out of 10 remaining homes with a security system; if the first home did not have a security system, then there are 8 out of 10 remaining homes with a security system.

The probability of all three of the selected homes having a security system is 0.3277, the probability of none of the three selected homes having a security system is 0.0414, and the probability of at least one of the selected homes having a security system is 0.9709. The events are dependent.

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The value of the range is 33

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

61, 61, 63, 64, 65, 66, 66, 66, 67, 68, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 94

To find the range of a set of numbers, we need to subtract the smallest number from the largest number.

In this set of numbers:

The smallest number is 61 and the largest number is 94.

Therefore, the range of this set of numbers is:

94 - 61 = 33

Hence, the range of the given set of numbers is 33.

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For answering 5 questions wrong Hunter's score is 75 and for for answering x question wrong his score will be  f(x)=100-5*x.

A function is defined as the relationship between input and output, where each input has exactly one output. The inputs are the elements in the domain and the outputs are elements in the co-domain.

Hunter is taking a multiple choice test with a total of 100 points available.

Each question is worth exactly 5 points,

From that the number of questions in the test is ,[tex]\frac{100}{5}=20[/tex]questions

Hunter's test score (out of 100) if he got 5 questions wrong,

20-5=15 questions correct.

Each question carries 5 marks, then 15*5= 75 marks.

Hunter score will be 75 marks.

And again each question carries 5 marks, If hunter answers x questions wrong his score will be f(x)=100-5*x.

It is a linear equation.

Hence, for answering 5 questions wrong Hunter's score is 75 and for for answering x question wrong his score will be f(x)=100-5*x.

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The Average Minute to Clean up a Car is 0.7794.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

The equation of the graph is y=0.7794x-1435.5.

When we compare the equation with the standard form of equation y=mx+b.

The m will be 0.7794 which means slope is 0.7794.

Hence,  0.7794 is the Average Minute to Clean up a Car.

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The surface area of the ramp is 3672 in².

Option A is the correct answer.

A rectangle is a two-dimensional shape where the length and width are different.

The area of a rectangle is given as:

Area = Length x width

We have,

The ramp consists of three rectangular surfaces and two triangles.

Now,

Triangle.

Base = 48 in

Height = 14 in

Area of a triangle.

= 1/2 x base x height

= 1/2 x 48 x 14

= 48 x 7

= 336 in²

Rectangle.

Since the back and the bottom rectangle are not included.

Front rectangle.

Length = 50 in

Width = 60 in

Area = 50 x 60

Area = 3000 in²

Now,

The surface area of the ramp.

= 300 + 2 (336)

= 3000 + 672

= 3672 in²

Thus,

The surface area of the ramp is 3672 in².

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The total amount spend is 391.88

The noun discount refers to an amount or percentage deducted from the normal selling price of something.

If 14 bag of mix cement are bought and and every third bag is discounted

no of bag on discount = 14/3 = 4 remainder 2

therefore 4 bags will be on discount

discount on a bag = 3.88× 75/100

= 291/100 = 2.91

therefore each bag on discount will cost = 3.88 - 2.91

= 0.97

for 4 bags = 0.97 ×4 = 3.88

for remaining 10 bags = 3.88 × 10

= 388

therefore the total amount = 388+3.88

= $391.88

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Answer:-17

Step-by-step explanation:

b=7,then

b^2=49

square root of 49-4*3*2=-17

The interest rate that can explain how $4081 grew to $8404.54 in 21 years with annual compounding is option (a) 3.5%, rounded to the nearest tenth.

The interest rate and the compounding frequency determine the growth of the initial amount over time. In this problem, we need to find the interest rate that can explain how an initial amount grew to a specific amount over a certain number of years.

The problem asks us to find the interest rate that can explain how an initial amount of $4081 grew to $8404.54 in 21 years with annual compounding. We can use the formula for compound interest to solve this problem:

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the future value of the investment, P is the initial amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, we know that P = $4081, A = $8404.54, n = 1 (annual compounding), and t = 21. We need to find the value of r that makes the equation true.

We can rearrange the formula to solve for r:

[tex]r = n[(A/P)^{1/nt} - 1][/tex]

Substituting the known values, we get:

[tex]r = 1[(8404.54/4081)^{1/(1*21)} - 1][/tex]

r = 0.0353 or 3.53%

This means that if the initial amount was invested with an annual interest rate of 3.5%, it would have grown to the given future value after 21 years.

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The system of equations has one solution, which is (2, 3).

To find the solution to a graphed system of equations, you need to find the points where the graphs of the two equations intersect. These points represent the coordinates of the solution, which are the values of x and y that make both equations true simultaneously.

The system of equations as shown in the graph has one solution, which is: (2, 3). This is because the liens intersect at the point (2, 3).

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Option A. No, this is not possible. Since A and B are not mutually exclusive events, we have P(A ∩ B) = P(A) + P(B) - P(A ∪ B), where P(A ∪ B) is the probability that the selected student has either a Visa or a MasterCard.

Since 0.5 > 1 - P(A ∪ B) = P(A') ∩ P(B'), we must have P(A') ∩ P(B') < 0, which is impossible.

B. The probability that the selected student has at least one of these two types of cards is given by P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.7 + 0.4 - 0.3 = 0.8.

C. The probability that the selected student has neither type of card is given by P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.8 = 0.2.

D. The probability of the event A ∩ B' is given by P(A ∩ B') = P(A) - P(A ∩ B) = 0.7 - 0.3 = 0.4.

E. The probability that the selected student has exactly one of the two types of cards is given by P((A ∩ B') ∪ (A' ∩ B)) = P(A ∩ B') + P(A' ∩ B) = 0.4 + 0.1 = 0.5.

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Answer:

first 2 pairs

Step-by-step explanation:

[tex]\frac{6}{100}[/tex] = 6 ÷ 100 = 0.06 ← equivalent

[tex]\frac{7}{10}[/tex] = 7 ÷ 10 = 0.7 ← equivalent

[tex]\frac{5}{10}[/tex] = 5 ÷ 10 = 0.5 ≠ 0.05

[tex]\frac{32}{100}[/tex] = 32 ÷ 100 = 0.32 ≠ 3.2

1. The ratio of the shaded area to the area of the whole figure is 3/8 and the ratio of the shaded area to the unshaded area is 3/5.

2. The value of AC/CD. CD/BD, and BD/BC is 8/5, 5/2 and 10/7 respectively.

A ratio is a comparison of two quantities, and it can be expressed in different ways, such as a fraction, a decimal, or a percentage. In this case, we will explore how to find ratios of areas in geometric figures.

Firstly, let's consider the rectangle on the right. To find the ratio of the shaded area to the area of the whole figure, we need to calculate both of these values. The shaded area is a rectangle with dimensions 6 by 2, which means its area is 6 x 2 = 12 square units.

The whole figure is a rectangle with dimensions 8 by 4, so its area is

=> 8 x 4 = 32 square units.

Therefore, the ratio of the shaded area to the area of the whole figure is 12/32, which simplifies to 3/8.

To find the ratio of the shaded area to the unshaded area, we need to subtract the shaded area from the whole area. The unshaded area is a rectangle with dimensions 8 by 2, so its area is

=> 8 x 2 = 16 square units.

Subtracting the shaded area from the whole area gives us

=> 32 - 12 = 20 square units for the unshaded area.

Thus, the ratio of the shaded area to the unshaded area is 12/20, which simplifies to 3/5.

Moving on to the second question, we need to use the figure below to find three ratios: AC/CD, CD/BD, and BD/BC.

Let's start with the first ratio, AC/CD. We know that AC is 8 units long and CD is 5 units long, so

=> AC/CD is 8/5.

Next, CD/BD. We know that BD is 10 units long, so to find CD/BD, we need to subtract the length of AC from the length of BD. AC is 8 units long, so

BD - AC = 10 - 8 = 2 units.

Therefore, CD/BD is 5/2.

Finally, BD/BC. We know that BC is 12 units long, so to find BD/BC, we need to subtract the length of CD from the length of BC. CD is 5 units long, so

=> BC - CD = 12 - 5 = 7 units.

Therefore, BD/BC is 10/7.

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The variable y does not vary directly with x is wrong since when x increases by 1 unit y increases by 1/2 unit for all real x.

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

here, we have,

Given that line passes through (0,0) and (2,1)

(Note that any two points are sufficient to determine the equation of the line)

The slope = change in y/change in x

                 = (1-0)/(2-0)

                 = 1/2

Hence answers are:

Variable y varies directly with x. (if x increases y also increases and vice versa)

The constant of variation = slope of line = 1/2

The constant of variation is 2 is wrong since slope = 1/2

The variable y does not vary directly with x is wrong since when x increases by 1 unit y increases by 1/2 unit for all real x.

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Answer:

y=67

z=113
x=67

Step-by-step explanation:

opposite angles are parallel in a rhombus and they all equal 360

Answer: Let's call the length of the cake "l" and the width of the cake "w". The total area of the cake would be l * w.

Half of the cake is frosted with vanilla icing, so the area of the vanilla half would be (l * w) / 2.

And 2/3 of the vanilla half is to be covered with sprinkles, so the area covered with sprinkles would be (l * w) / 2 * 2/3 = (l * w) / 3.

The other half of the cake is frosted with chocolate icing, so the area of the chocolate half would be (l * w) / 2.

And 1/4 of the chocolate half is to be covered with sprinkles, so the area covered with sprinkles would be (l * w) / 2 * 1/4 = (l * w) / 4.

So, the total area of the cake covered with sprinkles would be (l * w) / 3 + (l * w) / 4.

Step-by-step explanation:

The function [tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}[/tex] has a limit

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

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}[/tex]

Substitute 0 for x in the above expression

So, we have the following representation

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2} = \frac{1 - \sqrt[3]{0^2 + 1}}{0^2}[/tex]

Evaluate the expression

[tex]\lim_{x \to 0} \frac{1 - \sqrt[3]{x^2 + 1}}{x^2}=\infty[/tex]

This means that the limit of the expression is to infinity

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Answer:

it could be D but D gives us 661.63 C gives us 613.07 B gives us 892.165 and A gives us 883.98 so your closes answer is D

Step-by-step explanation:

8.42x109=917.78-253.8=663.98

The expected value of each grab bag is $1.67 for two pairs of lined gloves, $9.87 for one pair of lined gloves and one pair of knit gloves, and $16.80 for two pairs of knit gloves.  the total profit made from selling all 60 grab bags is $149.40.

For the 2 bags that have two pairs of lined gloves, the total cost is:

2 pairs * $25/pair = $50

Total gloves = 4

For the 16 bags that have one pair of lined gloves and one pair of knit gloves, the total cost is:

1 pair lined gloves * $25/pair + 1 pair knit gloves * $12/pair = $37

Total gloves = 4

For the 42 bags that have two pairs of knit gloves, the total cost is:

2 pairs * $12/pair = $24

Total gloves = 4

The probability of selecting a bag with two pairs of lined gloves is:

2/60 = 1/30

The probability of selecting a bag with one pair of lined gloves and one pair of knit gloves is:

16/60 = 4/15

The probability of selecting a bag with two pairs of knit gloves is:

42/60 = 7/10

The expected value of a bag with two pairs of lined gloves is:

(1/30) * $50 + (0) * $37 + (0) * $24 = $1.67

The expected value of a bag with one pair of lined gloves and one pair of knit gloves is:

(4/15) * $37 + (0) * $50 + (0) * $24 = $9.87

The expected value of a bag with two pairs of knit gloves is:

(7/10) * $24 + (0) * $50 + (0) * $37 = $16.80

The total cost of the gloves used to make the grab bags is:

2 bags * $50/bag + 16 bags * $37/bag + 42 bags * $24/bag = $1,650

The total revenue from selling all 60 grab bags is:

60 bags * $29.99/bag = $1,799.40

Therefore, the total profit is:

$1,799.40 - $1,650 = $149.40

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Answer:

[tex]x=\dfrac{1}{2},\;\;x=1,\;\;x=-\dfrac{4}{3}[/tex]

Step-by-step explanation:

Given polynomial function:

[tex]f(x) = 6x^3-x^2-9x+4[/tex]

If (2x - 1) is a factor of the given function, then 2x - 1 = 0.

Therefore:

[tex]\implies 2x-1=0[/tex]

[tex]\implies x=\dfrac{1}{2}[/tex]

To divide the given function by the factor (2x - 1), we can perform synthetic division.

As the leading coefficient of the divisor is not 1, divide the dividend and divisor by the leading coefficient of the divisor (2):

[tex]\textsf{Dividend}: \quad 3x^3-\dfrac{1}{2}x^2-\dfrac{9}{2}x+2[/tex]

[tex]\textsf{Divisor}: \quad \left(x-\dfrac{1}{2}\right)[/tex]

Perform Synthetic Division

Place ¹/₂ in the division box.

Write the coefficients of the dividend in descending order.

(Note: As no terms are missing, we do not need to use any zeros to fill in missing terms).

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\end{array}[/tex]

Bring the leading coefficient straight down:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}&\downarrow&&&\\\cline{2-5}&3\end{array}[/tex]

Multiply the number you brought down with the number in the division box and put the result in the next column (under the -¹/₂ ):

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&\frac{3}{2}&&\\\cline{2-5}&3\end{array}[/tex]

Add the two numbers together and put the result in the bottom row:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&&\\\cline{2-5}&3&1\end{array}[/tex]

Repeat:

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&\frac{1}{2}&\\\cline{2-5}&3&1&-4\end{array}[/tex]

[tex]\begin{array}{c|crrr}\vphantom{\dfrac12}\frac{1}{2} & 3 & -\frac{1}{2}& -\frac{9}{2} & 2\\\cline{1-1}\vphantom{\dfrac12}&\downarrow&-\frac{3}{2}&\frac{1}{2}&-2\\\cline{2-5}&3&1&-4&0\end{array}[/tex]

The bottom row (except the last number) gives the coefficients of the quotient. The degree of the quotient is one less than that of the dividend.

The last number in the bottom row is the remainder.

Therefore, the factored function after synthetic division is:

[tex]f(x)=(2x-1)(3x^2+x-4)[/tex]

The quadratic factor can be factored further:

[tex]\implies 3x^2+4x-3x-4[/tex]

[tex]\implies x(3x+4)-1(3x+4)[/tex]

[tex]\implies (x-1)(3x+4)[/tex]

Therefore, the fully factored function is:

[tex]f(x)=(2x-1)(x-1)(3x+4)[/tex]

To find the roots of the function, set each factor to zero and solve for x:

[tex]2x-1=0 \implies x=\dfrac{1}{2}[/tex]

[tex]x-1=0 \implies x=1[/tex]

[tex]3x+4=0 \implies x=-\dfrac{4}{3}[/tex]

Therefore, the roots of the given function are:

[tex]x=\dfrac{1}{2},\;\;x=1,\;\;x=-\dfrac{4}{3}[/tex]

Answer: The ratio $42/81$ can be simplified by dividing both the numerator and denominator by the greatest common factor of 42 and 81, which is 9.

$42/81 = (42 \div 9) / (81 \div 9) = 4/9$

So, the ratio $42/81$ can be expressed as the simplified fraction $\frac{4}{9}$.

Step-by-step explanation:

Answer:

[tex]x=-3,x=-3,x=7[/tex]

Step-by-step explanation:

[tex]We\ know\ that\ the\ roots\ of\ an\ equation\ are\ its\ intercepts\ cut\ on\ the\ x-axis.\\We\ can\ observe\ that\ the\ graph\ of\ y=f(x)\ cuts\ the\ x\ axis\ at\ the\ point (7,0)\\ and\ touches\ the\ x-axis\ at\ (-3,0).\\[/tex]

[tex]Hence,\\The\ curve\ represents\ a\ cubic\ polynomial\ with\ double\ roots\ (-3,0),(-3,0)\\ as\ well\ as\ a\ distinct\ root:\ (7,0)[/tex]

The measure of the angle B is found to be 184 degrees.

The angles given in question:

line AB || DC

∠B = 9x + 2

∠C = 5x - 4

So,

∠B + ∠C = 180 ( co interior angles )

9x + 2 + 5x - 4 = 180

9x  - 2 = 180

9x = 180 + 2

x = 182/9

Then,

∠B = 9x + 2

∠B = 9*182/9 + 2

∠B = 182 + 2

∠B = 184

Thus, the measure of the angle B is found to be 184 degrees.

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The diagram of the correct question is attached.

The inequality changes if both sides of the expressions are switched or multiplied by a negative sign

Inequalities are expressions not separated by an equal sign.

From the given inequalities

For -10.5 < 3e

3e > -10.5

e > -10.5/3

This shows that the sign changes

For b/6≤ 7

b ≤ 6(7)

b≤ 42

The inequality sign does not change.

For the inequality -4.5c > 9

4.5c < -9
c < -9/4.5

c < -2

The inequality sign changes.

For the inequality -21/5 ≤  -5/2 d

21/5 ≥ 5/2d

5/2d≤ 21/5

d ≤42/25

The inequality sign does not change

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