1. What is the relationship between AD and AB?
2. What is the relationship between ACD and BDC?
3. What is the relationship between AC and CB? Explain.
4. True or False? Because BD is the angle bisector of ABC, AB = CB​

First, we need to find the cross-sectional area of each figure:

Since the cross-sectional areas are equal at every level, we can set up the following equation:

5x = 9π

Solving for x, we get:

x = 9π/5

Now, we can find the volume of the rectangular prism:

V = 5x * 8 = 72π cubic meters

And the volume of the cylinder:

V = π(3^2) * 8 = 72π cubic meters

Since the volumes are equal, we can set up the following equation:

72π = 72π

Simplifying, we get:

9πx = 40π

Solving for x, we get:

x = 40/9

Rounding to the nearest tenth, we get:

x ≈ 4.4 meters

Therefore, the width of the rectangular prism is approximately 4.4 meters.

if Grandson Sinko puts the forks in groups of 12, there will be no forks left over

The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of each of the given integers. In other words, the LCM is the smallest number that is divisible by all the numbers in a given set.

the least common multiple of 4 and 6, is 12. So if Grandson Sinko puts the forks in groups of 12, there will be no forks left over.

To find the number of forks, we can use the fact that the number of forks is three more than a multiple of both 4 and 6. Let's call the number of forks x. Then we have:

x ≡ 3 (mod 4)

x ≡ 3 (mod 6)

Using the Chinese remainder theorem, we can combine these congruences to get:

x ≡ 3 (mod 12)

This means that the number of forks is three more than a multiple of 12. So we can write:

x = 12n + 3

where n is an integer. To find the value of n, we can plug in the values of x from the first two congruences:

x ≡ 3 (mod 4) => x = 4m + 3

x ≡ 3 (mod 6) => x = 6k + 3

Setting these two expressions for x equal to each other, we get:

4m + 3 = 6k + 3

Simplifying, we get:

2m = 3k

Since 2 and 3 are relatively prime, this means that m must be a multiple of 3 and k must be a multiple of 2. So we can write:

m = 3p

k = 2q

where p and q are integers. Plugging these into the expressions for x, we get:

x = 4m + 3 = 4(3p) + 3 = 12p + 3

x = 6k + 3 = 6(2q) + 3 = 12q + 3

Since x is the same in both expressions, we can set them equal to each other:

12p + 3 = 12q + 3

Simplifying, we get:

p = q

So we can write:

x = 12p + 3

where p is an integer. Therefore, if Grandson Sinko puts the forks in groups of 12, there will be no forks left over.

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The  five different math vocabulary that can be used to describe each given expression 6(t+2)-8

Expression: A combination of numbers, variables, and operators that can be evaluated or simplified.

Coefficient: A number that is multiplied by a variable.

Distributive Property: The property that allows you to multiply a factor by each term inside a set of parentheses.

Simplify: To reduce an expression to its simplest form.

Linear: An expression that has no exponents or other operations besides addition, subtraction, and multiplication by a constant.

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Step-by-step explanation:On a coordinate plane you move 9 to the right and 2 up

Answer:   To find the time at which the temperature is maximum, we need to find the vertex of the quadratic function T(x) = -0.07x^2. Recall that the x-coordinate of the vertex of a quadratic function f(x) = ax^2 + bx + c is given by -b/2a. In this case, a = -0.07 and b = 0 (since there is no linear term), so the x-coordinate of the vertex is x = -b/2a = -0/(-0.14) = 0.

Since x is the number of hours after 6 a.m., the time corresponding to x = 0 is 6 a.m. Therefore, the temperature is a maximum at 6 a.m.

To find the maximum temperature, we evaluate T(0) = -0.07(0)^2 = 0. Therefore, the maximum temperature is 0 degrees Fahrenheit. Note that this result makes sense, since the quadratic function T(x) = -0.07x^2 is a downward-facing parabola, which means that the temperature decreases as the number of hours after 6 a.m. increases.

Step-by-step explanation:

Answer: While I cannot give an exact answer without additional context, the answer should have a slope of 12, and should look like the image below.

Answer:

12

Step-by-step explanation:

Answer:

The factor by which the amount of drug in the body is multiplied by each passing hour can be found using the formula:

b = 1/2^(t/h)

where:

t = time elapsed (in hours)

h = half-life of the drug

In this case, the half-life of the drug is 5 hours, so we have:

h = 5 hours

For each passing hour, the time elapsed increases by 1 hour, so we can calculate the factor b after 1 hour as:

b = 1/2^1/5

b = 0.63

This means that after 1 hour, the amount of drug in the body is multiplied by 0.63. After 2 hours, it is multiplied by 0.63 again, and so on.

Answer:

[tex]v≈41033.52 \: {ft}^{3} [/tex]

Step-by-step explanation:

Given:

A sphere

r (radius) = 99 ft

π ≈ 3,14

Find: V (volume) - ?

[tex]v = \frac{4}{3} \pi {r}^{2} [/tex]

[tex]v = \frac{4}{3} \times 3.14 \times {99}^{2} ≈41033.52 \: {ft}^{3} [/tex]

Part (a)

Answers:

A' = (0, 2)

B' = (-1, -2)

C' = (-3, -2)

D' = (-4, -1)

Explanation:

Multiply each coordinate by 1/3.

For instance,

[tex]B(-3,-6) \to B( -3*(1/3), -6*(1/3) ) \to B \ '(-1, -2)[/tex]

====================================================

Part (b)

Answers:

A' = (-5, 5)

B' = (-8, -7)

C' = (-14, -7)

D' = (-17, -4)

Explanation:

Subtract 5 from the x coordinate. Subtract 1 from the y coordinate. This moves each point 5 units to the left and 1 unit down.

I'll show an example using point C.

[tex](\text{x},\text{y})\to(\text{x}-5,\text{y}-1)\\\\(-9,-6)\to(-9-5,-6-1)\\\\(-9,-6)\to(-14,-7)\\\\[/tex]

Therefore, C(-9,-6) moves to C ' (-14,-7)

Answer:

(G) -1

Step-by-step explanation:

Let the integer be [tex]x[/tex].

Then, it folloes that [tex]-x=x^2[/tex].

Because [tex]x \neq 0[/tex], dividing both sides by [tex]x>/tex] yields [tex]x=-1[/tex].

The leading coefficient of the polynomial is 4.

The leading coefficient of a polynomial, we first need to understand what the leading term of the polynomial is.

The leading term is the term with the highest degree in the polynomial.

For example, in the polynomial [tex]3x^4 + 2x^3 - 5x^2 + 4x - 1[tex], the leading term is 3x^4.
To determine the leading coefficient of a polynomial, we simply look at the coefficient of the leading term. For example, in the polynomial [tex]3x^4 + 2x^3 - 5x^2 + 4x - 1[/tex], the leading coefficient is 3.
Now, let's apply this concept to the polynomial graphed below. From the graph, we can see that the polynomial has a degree of 2, meaning that the highest power of x in the polynomial is 2.

Therefore, the leading term of the polynomial is [tex]ax^2[/tex], where a is the leading coefficient.
The vertex is the point on the graph where the polynomial reaches its maximum or minimum value. In this case, the vertex is located at [tex](-2, 4)[/tex]
Since the vertex is the highest point on the graph, we know that the coefficient of x^2 must be positive.

In this case, the value of y at the vertex is 4.

Therefore, the leading coefficient of the polynomial is 4.

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It asserts that if the sky is clear, then it is not possible for the plane to be on time.

In general, a hypothesis is a proposed explanation or tentative answer to a research question or problem. It is a statement that can be tested through investigation and analysis of data.

The statement "q -> ~p" can be interpreted as "If the sky is clear, then the plane is not on time."

This is a conditional statement, where q represents the hypothesis or condition ("the sky is clear") and ~p represents the conclusion or result ("the plane is not on time").

It asserts that if the sky is clear, then it is not possible for the plane to be on time.

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Thus, the surface area of the rectangular prism is found as : S = 275 ¹/₈ cm².

A rectangular prism is one of the more usual shapes for prisms. One definition of a rectangular prism is a prism with rectangle-shaped bases. An other faces also were rectangles because the bases are rectangles. These are a rectangular prism's characteristics:

Given dimensions for the rectangular prism :

Then ,

surface area of the prism S = 2(lw + wh + hl)

Put the values:

S = 2(11.75*5.75 + 5.75*4 + 4*11.75)

S = 2*137.5625

S = 275.125

S = 275 ¹/₈ cm²

Thus, the surface area of the rectangular prism is found as : S = 275 ¹/₈ cm².

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The average change in field position on each run was a loss of 7/4 yards.

A mixed number in mathematics is a number that combines a whole number and a fraction. The entire number and the fraction are often separated by a space or a plus sign when writing mixed numbers. As an illustration, the mixed number 2 1/3 stands for the sum of 2 and 1/3. By dividing the entire number by the fraction's denominator and adding the numerator, mixed numbers can be transformed into improper fractions. For instance, we would multiply 2 by 3 and add 1 to get 7, which is an incorrect fraction of 2 1/3. In order to obtain 7/3, we would then write the fraction using the same denominator as the original fraction.

The total change in position for a total loss of 15/3/4 yards is:

Total change in field position = - (15 + 3/4) yards

Given the, number of runs = 9

Thus:

Average change in field position = Total change in field position / Number of runs

Average change in field position = (- 15 - 3/4) yards / 9

Average change in field position = (- 63/4) yards / 9

Average change in field position = - 7/4 yards

Hence, the average change in field position on each run was a loss of 7/4 yards.

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8 possible sequences in total. Thus, A tree with 8 outcomes, option 1 is correct.

A diagram with a structure of branching connecting lines, representing different processes and relationships.

Here is a tree diagram showing all the possible outcomes of Sandra tossing a coin three times:

In the diagram, each branch represents the outcome of one toss of the coin. The letters "H" and "T" represent "heads" and "tails," respectively. Each path from the top of the tree to a bottom node represents a possible sequence of coin tosses, and there are 2³ = 8 possible sequences in total.

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Complete question:

Sandra tossed a coin 3 times. Which tree diagram shows al the possible outcomes of the coin landing heads up or tails up.

A tree with 8 outcomes

A tree with 12 outcomes

A tree with 9 outcomes

A tree with 6 outcomes

For given Sample Space, the probability that both responses are correct is 12.5%.

A sample space is the collection of all potential results of an experiment or random process in probability theory. It is a key idea that allows us to define and assess event probability.

Consider the following experiment: rolling a six-sided die. This experiment's sample space is the set of all potential results of rolling the dice, which are 1, 2, 3, 4, 5, 6. Each member of the sample space indicates a possible experiment outcome.

Now,

The probability of answering the true/false question correctly by random guess is 1/2, since there are two possible choices.

The probability of answering the multiple-choice question correctly by random guess is 1/4, since there are four possible choices.

The probability of both events happening together = product of individual probabilities:

P(correct on both) = P(true/false correct) × P(multiple-choice correct)

= (1/2) × (1/4)

= 1/8

= 0.125

Therefore, the probability that both responses are correct is 0.125 or 12.5%.

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Answer:20

Step-by-step explanation:

Answer:

[tex]-x^{5} -5x+6[/tex]

Step-by-step explanation:

Thus, the one notebook' price n = $6 and the price of one pencil is p = $2.4.

Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5. Once an equation is given in this format, finding both intercepts is rather simple (x and y). When attempting to solve systems involving two linear equations, this form is also quite helpful.

Let cost of a notebook to be 'n' dollars

Let the cost of a pencil to be 'p' dollars,

Then, the system of equations are-

3n + 5p = 30   ..eq 1

n + 10p = 30  ....eq2

multiply eq 2 with 3

3n + 30p = 90 ...eq 3

Subtract eq 3 from eq 1

3n + 5p - 3n - 30p = 30 - 90

25p = 60

p = 2.4

n + 10(2.4) = 30

n = 30 - 24

n = 6

Thus, the price of one notebook n = $6 and the price of one pencil is p = $2.4.

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Answer 6

Step-by-step explanation:  A = 1/2 bh, 6 x 2 is 12 and 1/2 of 12 is 6

Answer:

x=4

Step-by-step explanation:

We can solve for x by using the definition of logarithms.

log₃(x+5) = 2 can be rewritten in exponential form as 3² = x + 5.

Simplifying 3² gives 9, so we have:

x + 5 = 9

Subtracting 5 from both sides, we get:

x = 4

Therefore, the equivalent equation is x = 4.

Answer:
x > 53

30 + x > 83

Subtract 30 from both sides:

x > 53

Therefore, the solution is:

x > 53

Answer:

[tex]x = 1 \frac{49}{57} [/tex]

Step-by-step explanation:

[tex]6.75x + \frac{3}{8} x = 13 \frac{1}{4} [/tex]

Multiply the whole equation by 8, to eliminate the fractions:

[tex]54x + 3x = 106[/tex]

[tex]57x = 106[/tex]

Divide both parts of the equation by 57:

[tex]x = \frac{106}{57} = 1 \frac{49}{57} [/tex]

Answer:

y = 3 cot (1x)?

Step-by-step explanation:

The probability of picking a 7 and then picking a 7 is 8.33%.

Since we are not replacing the first card before drawing the second one, the probability of drawing a 7 on the first card is 1/4. After drawing the first card, there are three remaining cards, and only one of them is a 7. Therefore, the probability of drawing a 7 on the second card given that the first card was a 7 is 1/3.

Using the multiplication rule of probability, the probability of drawing a 7 on the first card and then drawing a 7 on the second card without replacement is

P(7 on the first card) x P(7 on the second card | 7 on the first card) = (1/4) x (1/3) = 1/12

P(7 on the first card and then a 7 on the second card) = 1/12 x 100% = 8.33%

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Triangle ABC's longest side is found as 4 units. Thus, Triangle DEF has a 100-times larger area than triangle ABC.

A ratio is a tool being used compare the sizes of two or more quantities with relation to one another in mathematics. By making amounts easier to understand, ratios enable us to evaluate and express quantities.

Given that both triangles remain similar, triangle DEF is 10 times larger than triangle ABC according to the ratio 1:10.

Thus, Triangle DEF has a 100-times larger area than triangle ABC.

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Answer:

Step-by-step explanation:

If 15 out of 50 students are twelve years old, we can estimate the proportion of twelve-year-olds in the whole school as follows:

Proportion of twelve-year-olds = 15/50

Simplifying this fraction by dividing both numerator and denominator by 5, we get:

Proportion of twelve-year-olds = 3/10

Similarly, if 20 out of 50 students are thirteen years old, we can estimate the proportion of thirteen-year-olds in the whole school as:

Proportion of thirteen-year-olds = 20/50

Simplifying this fraction by dividing both numerator and denominator by 10, we get:

Proportion of thirteen-year-olds = 2/5

Finally, if 15 out of 50 students are fourteen years old, we can estimate the proportion of fourteen-year-olds in the whole school as:

Proportion of fourteen-year-olds = 15/50

Simplifying this fraction by dividing both numerator and denominator by 5, we get:

Proportion of fourteen-year-olds = 3/10

Therefore, the best estimate of the proportion of students who are twelve, thirteen, and fourteen years old in the school, respectively, is:

3/10 are twelve years old

2/5 are thirteen years old

3/10 are fourteen years old

The probability of being in the first or second position in a heat with 10 cars can be calculated by finding the total number of ways that the cars can be arranged in the heat and then dividing by the number of ways that result in being in the first or second position.

There are 10 cars in the heat, so there are 10 possible positions for the first car, and once that car has been assigned a position, there are 9 possible positions remaining for the second car, and so on. Therefore, the total number of ways that the cars can be arranged is:

10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

This is equivalent to 10! (read as "10 factorial"), which is the product of all the positive integers from 1 to 10.

To be in the first or second position, there are 2 possible positions out of the 10 total positions in the heat. So, the number of ways to be in the first or second position is:

2 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

which is equivalent to 2 x 9!, since we only need to consider the arrangements of the remaining 9 cars after we have assigned ourselves to one of the first two positions.

Therefore, the probability of being in the first or second position is:

(2 x 9!) / 10!

Simplifying this expression, we get:

(2 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)

which simplifies further to:

2 / 10

or:

1 / 5

So, the probability of being in the first or second position is 1/5 or 0.2, which means that there is a 20% chance of being in one of those positions in any given heat.

Keep in mind that without the population standard deviation, it is impossible to provide an exact sample size. However, this formula will give you a good starting point.

To find the sample size needed for a 99.5% confidence interval with a margin of error of 1.5, we can use the formula:

n = (Z * σ / E)^2

where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.

For a 99.5% confidence interval, the Z-score is approximately 2.807 (from a standard normal distribution table). Since we do not have the population standard deviation (σ), we will need to estimate it using a sample standard deviation or use a conservative approach by assuming the maximum possible value. For now, let's assume we have an estimated standard deviation.

n = (2.807 * σ / 1.5)^2

Solve for n by plugging in the estimated standard deviation (σ) and then round up to the nearest whole number, as you cannot have a fraction of a sample.
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1. The two triangles are congruent, ASA criteria. angle H ≅ angle M, HY ≅ MN and EY ≅ AN using CPCT. 2. The two triangles are congruent, ASA criteria. AT ≅ AP, by CPCT. Angle T ≅ angle P. CT ≅ RP, by CPCT.

Triangle congruence: If all three corresponding sides and all three corresponding angles are equal in size, two triangles are said to be congruent. Slide, twist, flip, and turn these triangles to create an identical appearance. In mathematics, the term "congruent" refers to figures and shapes that can be flipped or rearranged to match up with other ones. These forms can be mirrored to produce related shapes.

If two shapes are similar in size and shape, they are congruent. We can also state that if two shapes are congruent, then their mirror images are identical.

1. For the given triangle HEY and triangle MAN we see that,

angle E ≅ angle A

angle Y ≅ angle N

and HE ≅ AM

Thus, the two triangles are congruent using the ASA criteria.

Also angle H ≅ angle M (CPCT)

HY ≅ MN

EY ≅ AN using CPCT.

2. For triangle CAT and PAR we have:

angle A ≅ angle A (vertically opposite angles)

Angle C ≅ angle R

AC ≅ AR

Thus, the two triangles are congruent using the ASA criteria.

Also, AT ≅ AP, by CPCT

Angle T ≅ angle P, by CPCT

CT ≅ RP, by CPCT.

3. For the triangles ACL and triangle ARS:

AC ≅ AR and angle 1 ≅ angle 2 (Given)

Also, angle A = angle A (Vertically opposite angle)

Thus, the two triangles are congruent using ASA criteria.

Now, angle 3 ≅ angle 4 using CPCT.

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1. The diagonal of the base of the box is shorter than the length of the wind chime. 2. The interior diagonal of the box shorter than the length of the wind chime. 3.  The wind chime will not fit in the box.

A diagonal is a line segment that connects two polygonal corners (vertexes), although it is not an edge (side). In other words, it connects any two polygonal vertices that are not neighboring. So, when we directly link any two vertices that are not connected by any sides, we may draw the diagonals in a polygon. A diagonal is a line that runs straight across the vertices of a polygon and joins its opposing corners. In other terms, a line segment connecting any two non-adjacent corners is a polygon's diagonal. Depending on how many sides a polygon has, distinct polygons may have a variable number of diagonals.

Given that the length of the wind chime is 16 inches thus,

1.  diagonal of base of the box = √10²+8² = √164 = 12.8

Hence, the diagonal of the base of the box is shorter than the length of the wind chime.

2.  interior diagonal of the box = √6²+8²+10² = √200 = 14.14

The interior diagonal of the box shorter than the length of the wind chime

3.  The wind chime will not fit in the box

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