Find the missing dimension of the triangle.

Answer:

(fg)(x) = x^4 - x^3 + 15x^2 - 6x + 54

Step-by-step explanation:

We want to multiply and simplify as much as possible:

f(x) * g(x)

(x^2 + 6)(x^2 - x + 9)

(x^2 * x^2) + (x^2 * - x) + (x^2 * 9) + (6 * x^2) + (6 * - x) + (6 * 9)

Note that when you're multiplying exponents, we add them:

x^4 + - x^3 + 9x^2 + 6x^2 - 6x + 54

Now we add 9x^2 and 6x^2 as they are like terms:

x^4 - x^3 + 15x^2 - 6x + 54

Thus, (fg)(x) simplified is x^4 - x^3 + 15x^2 - 6x + 54.

Optional:  Check the validity of the answer:

We can check that our answer is correct by plugging in a number for x in both the unsimplified and simplified expression and seeing if we get the same answer.  Let's try 5:

Plugging in 5 for x in (x^2 + 6)(x^2 - x + 9):

(5^2 + 6)(5^2 - 5 + 9)

(25 + 6)(25 - 5 + 9)

(31)(20 + 9)

(31)(29)

899

Plugging in 5 for x in x^4 - x^3 + 15x^2 - 6x + 54:

5^4 - (5)^3 + 15(5)^2 - 6(5) + 54

625 - 125 + 15(25) - 30 + 54

625 - 125 + 375 - 30 + 54

500 + 375 - 30 + 54

875 - 30 + 54

845 + 54

899

Thus, our answer is correct.

A counter-example to the given conclusion is the difference between the numbers 15 and 25, where both are positive, but the difference gives a negative result.

Here we are shown some differences between positive numbers:

7 - 3 = 4

8 - 5 = 3

9 - 8 = 1

And the conclusion is:

"The difference of two positive numbers is always positive".

To find a counter-example we just need to find two positive numbers such that the difference between these two positive numbers gives a negative number as a result.

An example of that can be the numbers 15 and 25, if we take the difference we will get:

15 - 25 = -10

Here we can see that the difference between two positive numbers gives a negative number as an outcome, so this is our counter-example.

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9514 1404 393

Answer:

  x = {-0.5-√1.25, -0.5+√1.25, 2, -0.5+i√0.75, -0.5-i√0.75}

Step-by-step explanation:

I like to use a graphing calculator to find clues as to the roots of higher-degree polynomials. Here, we see that x=2 is the only real rational root. Dividing that out by synthetic division, we see the remaining quartic factor is ...

  2x^5 -6x^3 -4x^2 -2x +4 = 0

  2(x -2)(x^4 +2x^3 +x^2 -1) = 0

We can recognize that the quartic factor is actually the difference of two squares:

  x^4 +2x^3 +x^2 -1 = (x^2 +x)^2 -1 = 0

So it resolves to two quadratic factors.

  (x^2 +x +1)(x^2 +x -1) = 0

One will have real roots, as shown by the graph. The other will have complex roots.

  x^2 +x + 1/4 = 1 +1/4 . . . . complete the square for the factor with real roots

  (x +1/2)^2 = 5/4

  x = -1/2 ± √(5/4) . . . . . . irrational real roots

__

  x^2 +x = -1 . . . . . . . . . . the quadratic factor with complex roots

  (x +1/2)^2 = -1 +1/4 . . . complete the square

  x = -1/2 ± i√(3/4) . . . . irrational complex roots

__

In summary, the values of x that satisfy the equation are ...

 x = 2

  x = -1/2 ± √(5/4)

  x = -1/2 ± i√(3/4)

Answer:

\(g'(y)=3y^2-4y-8\)

Step-by-step explanation:

start by foiling out the given function

\(g(y)=(y-4)(2y+y^2)\\=2y^2+y^3-8y-4y^2\\=y^3-2y^2-8y\)

next, use the power rule to find the derivative

power rule: To use the power rule, multiply the variable's exponent n, by its coefficient a, then subtract 1 from the exponent. If there's no coefficient (the coefficient is 1), then the exponent will become the new coefficient.

\(g'(y)=3y^2-4y-8\)

Answer:

L = 9

Step-by-step explanation:

6x8=48

432÷48=9

Answer:

Just some background:

Congruent means that a triangle has the same angle measures and side lengths of another triangle.

SAS congruence theorem: If two sides and the angle between these two sides are congruent to the corresponding sides and angle of another triangle, then the two triangles are congruent. Congruent triangles: When two triangles have the same shape and size, they are congruent.

Let's look at A first.

The triangle on left, we know 40 and 30 degree angles. So 3 all angles together = 180, so the 3rd angle = 180-30-40 = 110.

Now look at the triangle on the right. The angle shown is 110! This angle is between the sides marked with || and ||| marks, indicating that those two sides are the same length between both triangles.

Therefore both triangles are the same by Side-Angle-Side or SAS.

Now look at B.

It's a right triangle. We are missing 1 side of each triangle.

Let's solve for the missing "leg" of the triangle on the right. The pythagorean theorem says that a^2 + b^2 = c^2 where a and b are the 'legs' or sides of the triangle and c is the hypotenuse (always the longest length opposite the right angle).

so 2^2 + b^2 = 4^2

4 + b^2 = 16

b^2 = 16-4

b^2 = 12

That missing side is the \(\sqrt{12}\).

This does NOT match the triangle on the left.

Theses two triangles are NOT congruent.

Answer: The amount you would pay (including tax) for the item is:

$755.37 + $52.88 = $808.25

Step-by-step explanation: irst, we calculate the amount of discount that is applied to the original price:

30% of $1199 = 0.3 x 1199 = $359.70

So, the discounted price is:

$1199 - $359.70 = $839.30

Next, we apply the additional 10% off coupon to this discounted price:

10% of $839.30 = 0.1 x $839.30 = $83.93

The final price after the coupon is applied is:

$839.30 - $83.93 = $755.37

Finally, we calculate the sales tax on this final price:

7% of $755.37 = 0.07 x $755.37 = $52.88

The amount you would pay (including tax) for the item is:

$755.37 + $52.88 = $808.25

The correct answer is option D: A line passes through the point (0, 0) and continues through the point (12, 3)

For the first question:

The table shows a proportional relationship between x and y. To describe what the graph of this proportional relationship would look like, we can examine the given data points.

The data points are (12, 3), (8, 26), and (24, ?). We can observe that as x increases, y also increases. This indicates a positive correlation between the variables. Additionally, we can see that the ratio of y to x remains constant for each data point.

Now, let's analyze the answer options:

A. A line passes through the point (0, 0) and continues through the point (3, 12).

This answer option does not align with the given data because there is no data point with an x-value of 3 and a corresponding y-value of 12.

B. A line passes through the point (0, 0) and continues through the point (2, 8).

This answer option aligns with the given data because (2, 8) is a valid data point from the table, and the line passes through the origin (0, 0).

C. A line passes through the point (0, 0) and continues through the point (6, 24).

This answer option does not align with the given data because there is no data point with an x-value of 6 and a corresponding y-value of 24.

D. A line passes through the point (0, 0) and continues through the point (12, 3).

This answer option aligns with the given data because (12, 3) is a valid data point from the table, and the line passes through the origin (0, 0).

Based on the analysis, the correct answer is option D: A line passes through the point (0, 0) and continues through the point (12, 3). This accurately represents the graph of the proportional relationship given in the table.

For the second question, I'm sorry, but you didn't provide the multiple-choice options or the complete question. Could you please provide the question and options so that I can assist you further?

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Answer: d.12

Step-by-step explanation:

Answer:

he values of F(x) between x=1 and x=3 are 1, 4, and 9.

Step-by-step explanation:

To find the value of F(x) between x=1 and x=3, we need to substitute the values of x into the given function F(x) = x^2 and evaluate the function at each value. This gives us:

F(1) = 1^2 = 1

F(2) = 2^2 = 4

F(3) = 3^2 = 9

Therefore, the values of F(x) between x=1 and x=3 are 1, 4, and 9.

Based on their given probabilities, e would expect the color blue or green to be spun approximately 75 times in this experiment.

To find the expected number of times the color blue or green will be spun, we first need to add their probabilities:

0.25 + 0.125 = 0.375

This means that, on average, we would expect to see blue or green spun 0.375 times for every spin. To find the expected number of times this will happen over 200 spins, we can multiply the probability by the number of spins:

0.375 x 200 = 75

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The total weight of Rice that Mark buys is given as follows:

2.5 pounds.

The total weight of Rice that Mark buys is obtained applying the proportions in the context of the problem.

The weight of each bag is given as follows:

5/8 pounds = 0.625 pounds.

The number of bags is given as follows:

4 bags.

Hence the total weight of Rice that Mark buys is given as follows:

4 x 0.625 = 2.5 pounds.

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Answer:

314.16 in²

Step-by-step explanation:

Answer:

\(A\) ≈ \(314.16\)

Step-by-step explanation:

\(A=\pi r^2\)

\(A = \pi (10)^2\)

\(A = \pi (100)\)

\(A\) ≈ \(314.16\)

Answer:

(a)\(e^{-0.000428 t}\)

Step-by-step explanation:

We are given that

Half life of radium-226=1620 yr

The quantity left Q(t) after t years is given by

\(Q(t)=(\frac{1}{2})^{\frac{t}{1620}}\)

a. We have to convert the given function into an exponential function using base e.

\(Q(t)=(\frac{1}{2})^{\frac{t}{1620}}\)

=\(((\frac{1}{2})^t)^{\frac{1}{1620}\)

=\(e^{ln(1/2) t/1620}\)

\(=e^({\frac{ln(1/2)}{1620}t)\)

=\(e^{-0.000428 t}\)

(b)

\(Q(0)=e^{-0.000428 \times 0}\)

=1

From original function

Q(0)=1

\(Q(1620)=(\frac{1}{2})^{\frac{t}{1620}}\)

\(Q(1620)=\frac{1}{2}=0.5\)

From exponential function

\(Q(1620)=e^{-0.000428 \times 1620}\)

\(=0.499\approx 0.5\)

\(Q(3240)=(\frac{1}{2})^{\frac{3240}{1620})=0.25\)

\(Q(3240)=e^{-0.000428 \times 3240}\)

Q(3240)=0.249=\(\approx 0.25\)

Hence, verified.

Answer:

ALGUIEN PARA COJER?

TENGO 20 PERRO

Answer:

c

Step-by-step explanation:

Answer:

5.14 mi/h

Step-by-step explanation:

( sorry if wrong )

Answer:

5.6

Step-by-step explanation:

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

= 56 / 10

= 5.6

Answer:

$49.28 is the sale cost.

Step-by-step explanation:

0.35 x 75.82 = 26.537

75.82 - 26.537

= 49.283 and round

The selling cost of the book will be equal to $49.28.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that Judy is looking to buy a textbook. The book normally costs $75.82, but it is on sale for 35% below the regular cost.

The selling cost of the book will be calculated as,

Selling Cost = 0.35 x 75.82 = 26.537

Selling Cost  = 75.82 - 26.537

Selling Cost  = 49.283

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Answer:

a) The probability that in​ tonight's game the basketball player misses for the first time on his sixth attempt is 0.0311 = 3.11%.

b) The probability that in​ tonight's game the basketball player makes his first basket on his fifth shot is 0.0154 = 1.54%.

c) The probability that in​ tonight's game the basketball player makes his first basket on one of his first 3 shots is 0.936 = 93.6%.

Step-by-step explanation:

For each shot, there are only two possible outcomes. Either the player makes it, or he does not. The probability of making a shot is independent of other shots. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

In which \(C_{n,x}\) is the number of different combinations of x objects from a set of n elements, given by the following formula.

\(C_{n,x} = \frac{n!}{x!(n-x)!}\)

And p is the probability of X happening.

A basketball player has made 60​% of his foul shots during the season.

This means that \(p = 0.6\)

a) Misses for the first time on his sixth attempt

Makes the first five, which is P(X = 5) when n = 5.

Misses the sixth, with probability = 1-0.6 = 0.4.

So

\(p = 0.4P(X = 5)\)

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(p = 0.4P(X = 5) = 0.4*(C_{5,5}.(0.6)^{5}.(0.4)^{0}) = 0.0311\)

The probability that in​ tonight's game the basketball player misses for the first time on his sixth attempt is 0.0311 = 3.11%.

b) Makes his first basket on his fifth shot

Misses the first four, which is P(X = 0) when n = 4.

Makes the fifth, with a probability of 0.6.

So

So

\(p = 0.6P(X = 0)\)

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(p = 0.6P(X = 0) = 0.6*(C_{4,0}.(0.6)^{0}.(0.4)^{4}) = 0.0154\)

The probability that in​ tonight's game the basketball player makes his first basket on his fifth shot is 0.0154 = 1.54%.

c) Makes his first basket on one of his first 3 shots

Either he makes his first basket on one of his first 3 shots, or he misses all of them. The sum of these probabilities is decimal 1.

Misses the first three:

P(X = 0) when n = 3. So

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(P(X = 0) = C_{3,0}.(0.6)^{0}.(0.4)^{3} = 0.064\)

Makes on one of his first three:

1 - 0.064 = 0.936

The probability that in​ tonight's game the basketball player makes his first basket on one of his first 3 shots is 0.936 = 93.6%.

Answer:

Step-by-step explanation:

The probability that the basket player made a foul shot is 60% which is 0.60

Then the probability of good shot is 1 - 0.60 = 0.40

P = 0.40

a) the probability that the basket player misses for the first time on his sixth attempt ​is

P (first time on his sixth attempt) = (1 - P)⁵ (P)

= (1 - 0.4)⁵(0.4)

= (0.6)⁵(0.4)

= 0.07776 * 0.4

= 0.031104

≅ 0.0311

The probability that​  the basketball player misses for the first time on his sixth attempt is 0.0311

b) P(first basket on his fifth shot) = (1 - P)³ (P)

= (1 - 0.4)⁴(0.4)

= (0.60)⁴(0.4)

= 0.0518

c) The probability of making his first basket in first shot is 0.6

and the  probability of making his first basket in second shot is

0.6 * 0.4 = 0.24

the  probability of making his first basket in third shot is

0.6 * 0.4² = 0.096

So, the probability that the player makes his first basket on one of his first 3 shots is

= 0.6 + 0.24 + 0.096

= 0.936

Thus, the probability that in​ tonight's game the basketball player makes his first basket on one of his first 3 shots is 0.936

The measure of distance x, that is, the distance between a point P and the point of horizon, is equal to 284.372 miles.

In this problem we must determine the distance between a point P located about earth's circumference and the point of horizon, located on earth's circumference. Since the line between these two points is tangent to earth, then, distance x can be found by Pythagorean theorem:

x = √[(3959 mi + 10.2 mi)² - (3959 mi)²]

x = 284.372 mi

The distance x is equal to 284.372 miles.

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The domain of (boa)(x) is [1, ∞].

In Mathematics and Geometry, a domain is the set of all real numbers (x-values) for which a particular equation or function is defined.

Based on the information provided above, we have the following functions:

a(x) = 3x+1

\(b(x) = \sqrt{x-4}\)

Therefore, the composite function (boa)(x) is given by;

\(b(x) = \sqrt{3x+1 -4}\\\\b(x) = \sqrt{3x-3}\)

By critically observing the graph shown in the image attached below, we can logically deduce the following domain:

Domain = [1, ∞] or {x|x ≥ 1}.

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Answer:

\(\frac{d}{dx}[f(x)+g(x)+h(x)] = \frac{9\cdot x^{8}}{\sqrt{1-x^{18}}} - 81\cdot x^{80}-2\cdot x\)

Step-by-step explanation:

This derivative consist in the sum of three functions: \(f(x) = 81\cdot \sin^{-1} x^{9}\), \(g(x) = - x^{81}\) and \(h(x) = - x^{2}\). According to differentiation rules, the derivative of a sum of functions is the same as the sum of the derivatives of each function. That is:

\(\frac{d}{dx} [f(x)+g(x) + h(x)] = \frac{d}{dx} [f(x)]+\frac{d}{dx} [g(x)] +\frac{d}{dx} [h(x)]\)

Now, each derivative is found by applying the derivative rules when appropriate:

\(f(x) = 81\cdot \sin^{-1} x^{9}\) Given

\(f'(x) = \frac{9\cdot x^{8}}{\sqrt{1-x^{18}}}\) (Derivative of a arcsine function/Chain rule)

\(g(x) = - x^{81}\) Given

\(g'(x) = -81\cdot x^{80}\) (Derivative of a power function)

\(h(x) = - x^{2}\) Given

\(h'(x) = -2\cdot x\) (Derivative of a power function)

\(\frac{d}{dx}[f(x)+g(x)+h(x)] = \frac{9\cdot x^{8}}{\sqrt{1-x^{18}}} - 81\cdot x^{80}-2\cdot x\) (Derivative for a sum of functions/Result)

The minimum number of monthly checks Regina needs to write for the first account to be a better option is 21 or more. Answer C. 21 is one of the suggested choices.

On the basis of the quantity of checks written, we can compare the overall costs of the two accounts.

Let's calculate the total cost for each account based on the given information:

First Account:

Monthly fee: $12

Per-check fee: $0.10

Total cost = Monthly fee + (Per-check fee * Number of checks)

Second Account:

Monthly fee: $9

Per-check fee: $0.25

Total cost = Monthly fee + (Per-check fee * Number of checks)

We must determine the point at which the first account's total cost is less than the second account's total cost. Let's construct the formula:

$12 + ($0.10 * Number of checks) < $9 + ($0.25 * Number of checks)

Simplifying the equation

$12 - $9 < ($0.25 - $0.10) * Number of checks

$3 < $0.15 * Number of checks

Dividing both sides by $0.15:

$3 / $0.15 < Number of checks

20 < Number of checks

Therefore, the minimum number of monthly checks Regina needs to write for the first account to be a better option is 21 or more. Answer C. 21 is one of the suggested choices.

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Given statement solution is :- The actual length of the part is 3751/64 inches.

To find the length of the actual part, you can use the scale factor and the length of the part on the drawing. The formula for finding the actual length is:

Actual Length = Length on Drawing × Scale Factor

In this case, the length on the drawing is given as 3 7/8 inches, and the scale factor is given as 15 ¹/8. Let's calculate the actual length:

Length on Drawing = 3 7/8 inches = (3 × 8 + 7) / 8 = 31/8 inches

Scale Factor = 15 ¹/8

Now we can substitute the values into the formula:

Actual Length = (31/8 inches) × (15 ¹/8)

To perform the multiplication, we can convert the mixed fraction into an improper fraction:

15 ¹/8 = (15 × 8 + 1) / 8 = 121/8

Now we can multiply the fractions:

Actual Length = (31/8) × (121/8)

To multiply fractions, we multiply the numerators together and the denominators together:

Actual Length = (31 × 121) / (8 × 8)

Actual Length = 3751 / 64

Therefore, the actual length of the part is 3751/64 inches.

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Answer:

1) B = 88°, a = 13.1, b = 26.9

2) C = 73°, a = 20.9, b = 12.6

Step-by-step explanation:

To solve for the remaining sides and angles of the triangle, given two sides and an adjacent angle, use the Law of Sines formula:

\(\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}\)

Given values:

As the interior angles of a triangle sum to 180°:

\(\implies A+B+C=180^{\circ}\)

\(\implies B=180^{\circ}-A-C\)

\(\implies B=180^{\circ}-29^{\circ}-63^{\circ}\)

\(\implies B=88^{\circ}\)

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

\(\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)

\(\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}\)

Solve for a:

\(\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{24}{\sin 63^{\circ}}\)

\(\implies a=\dfrac{24\sin 29^{\circ}}{\sin 63^{\circ}}\)

\(\implies a=13.0876493...\)

\(\implies a=13.1\)

Solve for b:

\(\implies \dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}\)

\(\implies b=\dfrac{24\sin 88^{\circ}}{\sin 63^{\circ}}\)

\(\implies b=26.9194211...\)

\(\implies b=26.9\)

\(\hrulefill\)

Given values:

As the interior angles of a triangle sum to 180°:

\(\implies A+B+C=180^{\circ}\)

\(\implies C=180^{\circ}-A-B\)

\(\implies C=180^{\circ}-72^{\circ}-35^{\circ}\)

\(\implies C=73^{\circ}\)

Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:

\(\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)

\(\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}\)

Solve for a:

\(\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{21}{\sin 73^{\circ}}\)

\(\implies a=\dfrac{21\sin 72^{\circ}}{\sin 73^{\circ}}\)

\(\implies a=20.8847511...\)

\(\implies a=20.9\)

Solve for b:

\(\implies \dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}\)

\(\implies b=\dfrac{21\sin 35^{\circ}}{\sin 73^{\circ}}\)

\(\implies b=12.5954671...\)

\(\implies b=12.6\)

The value of the coin in 2019 would be approximately $132.

To calculate the value of the coin in 2019, we need to consider the annual increase rate of 16.9% from 2010 to 2019. We can use the compound interest formula to find the final value.

Starting with the initial value of $40 in 2010, we can calculate the value in 2019 as follows:

Value in 2019 = Initial value * (1 + Rate)^n

where Rate is the annual increase rate and n is the number of years between 2010 and 2019.

Plugging in the values:

Value in 2019 = $40 * (1 + 0.169)^9

Value in 2019 ≈ $40 * 2.996

Value in 2019 ≈ $119.84

Rounding the value to the nearest dollar, we get approximately $120. Therefore, the value of the coin in 2019 would be approximately $120.

However, please note that the exact value may vary depending on the specific compounding method and rounding conventions used.

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