The problem is to find the volume of region E enclosed by a cone and a sphere. The solution involves converting the equations to cylindrical coordinates, finding the limits of integration, and setting up a triple integral. The volume can be calculated by evaluating the integral.
To compute the volume of E using cylindrical coordinates, we first need to find the limits of integration for r, θ, and z. Since E is enclosed by the cone 7 = — x² + y² and the sphere x² + y2 + z2 = 32, we need to find the equations that define the boundaries of E in cylindrical coordinates.
To do this, we convert the equations of the cone and sphere to cylindrical coordinates:
- Cone: 7 = — x² + y² → 7 = — r² sin² θ + r² cos² θ → r² = 7 / sin² θ
- Sphere: x² + y² + z² = 32 → r² + z² = 32
We can see that the cone intersects the sphere when r² = 7 / sin² θ and r² + z² = 32. Solving for z, we get z = ±√(32 - 7/sin² θ - r²). We also know that the cone extends to the origin (r = 0), so our limits of integration for r are 0 to √(7/sin² θ).
For θ, we can see that E is symmetric about the z-axis, so we can integrate over the entire range of θ, which is 0 to 2π.
For z, we need to find the range of z values that are enclosed by the cone and sphere. We can see that the cone intersects the z-axis at z = ±√7. We also know that the sphere intersects the z-axis at z = ±√(32 - r²). Thus, the range of z values that are enclosed by the cone and sphere is from -√(32 - r²) to √(32 - r²) if r < √7, and from -√(32 - 7/sin² θ) to √(32 - 7/sin² θ) if r ≥ √7.
Now that we have our limits of integration, we can set up the triple integral to compute the volume of E:
Vol(E) = ∫∫∫ E dV
= ∫₀^(2π) ∫₀^√(7/sin² θ) ∫₋√(32 - r²)^(√(32 - r²)) F(r, θ, z) dz dr dθ
where F(r, θ, z) = 1 (since we're just computing the volume of E).
Using the limits of integration we found, we can evaluate this triple integral using numerical integration techniques or a computer algebra system.
To find the volume of the region E enclosed by the cone 7 = -x² + y² and the sphere x² + y² + z² = 32, we can use triple integration in cylindrical coordinates. We need to determine the limits of integration for r, θ, and z.
First, rewrite the equations in cylindrical coordinates:
Cone: z = -r² + 7
Sphere: r² + z² = 32
Now, find the intersection between the cone and the sphere by solving for z in the cone equation and substituting it into the sphere equation:
r² + (-r² + 7)² = 32
Solving for r, we get r = √7.
Now, we can find the limits of integration:
r: 0 to √7
θ: 0 to 2π
z: -r² + 7 to √(32 - r²)
Since the volume is the region enclosed by these surfaces, we can set up the triple integral:
Vol(E) = ∫∫∫ r dz dθ dr
With the limits of integration:
Vol(E) = ∫(0 to 2π) ∫(0 to √7) ∫(-r² + 7 to √(32 - r²)) r dz dθ dr
Evaluating this integral will give us the volume of the region E.
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Solve each system by substitution
Y=-2x+4
y=-3x+3
Answer: x = -1, y = 6
Step-by-step explanation:
lets substitute the value of y from the first equation into the y in the second equation.
-2x + 4 = -3x + 3
4 - 3 = -3x + 2x
1 = -1x
x = -1
we know from before that y = -2x + 4
so y = -2(-1) + 4
y = 6
Find the t value that forms the boundary of the critical region in the right-hand tail for a one-tailded test with o=. 01 for each of the folling sample size n=10
The t critical value at 29 degrees of freedom and 0.01 level of significance is 2. 46
How to calculate the valueUsing Critical value calculator we calculate the values.
a) at n = 10
Therefore degrees of freedom is = n - 1= 9, So therefore at 9 degrees of freedom and 0.01 level of significance, t critical value is 2.82
b) at n= 20
Degrees of freedom is 19.
The t critical value at 19 degrees of freedom and 0.01 level of significance is 2.54
c) at n = 30
Degrees of freedom is 29.
So therefore t critical value at 29 degrees of freedom and 0.01 level of significance is 2. 46
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Find f'(4) for f(x) = ln (2x^3"). Answer as an exact fraction or round to at least 2 decimal places.
To find f'(4) for the function f(x) = ln(2x^3), we first need to find the derivative f'(x) using the chain rule.
The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
For f(x) = ln(2x^3), the outer function is ln(u) and the inner function is u = 2x^3.
The derivative of the outer function, ln(u), is 1/u.
The derivative of the inner function, 2x^3, is 6x^2 (using the power rule).
Now, apply the chain rule: f'(x) = (1/u) * 6x^2 = (1/(2x^3)) * 6x^2.
Simplify f'(x): f'(x) = 6x^2 / (2x^3) = 3/x.
Now, find f'(4): f'(4) = 3/4.
So, f'(4) for f(x) = ln(2x^3) is 3/4 or 0.75 when rounded to 2 decimal places.
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Use matrices A, B, C and D. A = Find CD a. 2 3 0 -5 b. 27 -29 -7 31.0 -384-1 2 6 -6 -6 4 -2 -27 18 -9 -12 8 -4 2 Mark this and return C = 9 and D= [-3 2 -1] C. Please select the best answer from the choices provided | ** - 16 24 -8 0 -40 32 Save and Evil
The product of matrices C and D is:
[tex]CD = \left[\begin{array}{ccc}-6&18&-4\\\end{array}\right][/tex]
The best answer is option b.
How to find the product of two matrices?A matrix (plural matrices) is a set of numbers arranged in rows and columns so as to form a rectangular array.
The number of rows of a matrix can be determined by counting from top to bottom and the number of columns can be determined by counting from left to right.
The product (multiplication) of matrices C and D is:
CD = C * D
[tex]CD = \left[\begin{array}{ccc}2\\9\\4\end{array}\right] * \left[\begin{array}{ccc}-3&2&-1\\\end{array}\right][/tex]
To get the product, multiply each row by the column. That is:
2 * (-3) = -6
9 * 2 = 18
4 * (-1) = -4
[tex]CD = \left[\begin{array}{ccc}-6&18&-4\\\end{array}\right][/tex]
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What is the mean absolute deviation of 10 10 9 8 10 5 6 4 8 4
The mean absolute deviation of the given data set is approximately 2.16.
To find the mean absolute deviation (MAD), we first need to calculate the mean of the data set:
Mean = (10 + 10 + 9 + 8 + 10 + 5 + 6 + 4 + 8 + 4) / 10 = 7.4
Next, we calculate the absolute deviation of each data point from the mean:
|10 - 7.4| = 2.6
|10 - 7.4| = 2.6
|9 - 7.4| = 1.6
|8 - 7.4| = 0.6
|10 - 7.4| = 2.6
|5 - 7.4| = 2.4
|6 - 7.4| = 1.4
|4 - 7.4| = 3.4
|8 - 7.4| = 0.6
|4 - 7.4| = 3.4
Then, we find the average of these absolute deviations:
MAD = (2.6 + 2.6 + 1.6 + 0.6 + 2.6 + 2.4 + 1.4 + 3.4 + 0.6 + 3.4) / 10 ≈ 2.16
Therefore, the mean absolute deviation of the given data set is approximately 2.16.
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Flo ate
3
2
of a sandwich and Arnie ate- of a sandwich. If Arnie ate more, what
3
must be true?
A Flo's sandwich is bigger.
B Arnie's sandwich is bigger.
C) The sandwiches are the same size.
D) It doesn't matter which sandwich is bigger.
Flo ate more of the sandwich than Arnie.
Option A is the correct answer.
We have,
We need to compare the values 3/4 and 2/3 to determine which fraction represents a larger amount of sandwiches eaten.
To make the fractions comparable, we need to find a common denominator.
The least common multiple of 4 and 3 is 12.
So we can rewrite 3/4 and 2/3 with 12 as the denominator:
3/4 = 9/12
2/3 = 8/12
Comparing these fractions, we see that 9/12 (or 3/4) is greater than 8/12
(or 2/3).
Therefore,
Flo ate more of the sandwich than Arnie.
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A license plate is made of three letters and three numbers, how many different license plates are possible?
There are 17,576,000 different license plates possible, considering 26 letters (A-Z) and 10 numbers (0-9).
There are 26 options for each of the three letters and 10 options for each of the three numbers. Therefore, using the multiplication principle, the total number of possible license plates is 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000.
Alternatively, we can use the permutation formula to calculate the number of arrangements: P(26,3) x P(10,3) = 15,600 x 720 = 11,251,200.
However, since order does not matter in a license plate, we need to divide by the number of permutations of three letters and three numbers, which is 3! x 3! = 36, resulting in 11,251,200 / 36 = 17,576,000 possible license plates.
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A peregrine falcon can dive at the speed of 320km/h. Create a problem that you can solve by finding an equivalent rate for this speed. Then solve the problem.
pls help <3 Triangle QRS has side lengths q = 11, r = 17, and s = 23. What is the measure of angle R
a.44.5°
b.59.3°
c.27.0°
d.108.6
Using the cosine law, the measure of angle R is calculated as approximately: a. 44.5°.
How to Use the Cosine Law to Solve a Triangle?The cosine law is expressed as follows:
cos R = [s² + q² – r²]/2sq
Given the following side lengths of triangle QRS:
Side q = 11,
Side r = 17,
Side s = 23.
Plug in the values into the cosine law formula:
cos R = [23² + 11² – 17²]/2 * 23 * 11
cos R = 361/506
Cos R = 0.7134
R = cos^(-1)(0.7134)
R ≈ 44.5°
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Find an angle in each quadrant with a common reference angle with 306°, from 0°≤θ<360°
The angle in each quadrant with a common reference angle with 306° are
Quadrant 1 is 126°Quadrant 2 is 54°Quadrant 3 is 306°Quadrant 4 is 234°To find angles in each quadrant with a common reference angle with 306°, we first need to determine the reference angle for 306°.
Reference angle is the acute angle between the terminal side of an angle and the x-axis. We can find the reference angle for any angle θ by subtracting the nearest multiple of 180° from θ and taking the absolute value of the result. In this case:
|306° - 180°| = 126°
So, the reference angle for 306° is 126°.
Now, we can find an angle in each quadrant with a common reference angle of 126°:
1st quadrant: The angle with a reference angle of 126° in the 1st quadrant is simply 126°.
2nd quadrant: To find the angle with a reference angle of 126° in the 2nd quadrant, we need to subtract the reference angle from 180° (since all angles in the 2nd quadrant are between 90° and 180°).
180° - 126° = 54°
So, an angle with a reference angle of 126° in the 2nd quadrant is 54°.
3rd quadrant: To find the angle with a reference angle of 126° in the 3rd quadrant, we need to subtract the reference angle from 180° and then add 180° (since all angles in the 3rd quadrant are between 180° and 270°).
180° + 126° = 306°
So, an angle with a reference angle of 126° in the 3rd quadrant is 306°.
4th quadrant: To find the angle with a reference angle of 126° in the 4th quadrant, we need to subtract the reference angle from 360° (since all angles in the 4th quadrant are between 270° and 360°).
360° - 126° = 234°
So, an angle with a reference angle of 126° in the 4th quadrant is 234°.
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Which of these could be the side lengths of a right triangle? list all possible answers and show your work for full marks.
a) 4-7-10
b) 36-48-60
c) 6-10-14
d) 14-48-50
The sets of side lengths that could form a right triangle are 36-48-60 (option b) and 14-48-50 (option d).
To determine which of these sets of side lengths could form a right triangle, we will use the Pythagorean theorem (a² + b² = c²), where a and b are the shorter sides and c is the hypotenuse. Let's evaluate each option:
a) 4-7-10
Applying the Pythagorean theorem: 4² + 7² = 16 + 49 = 65, which is not equal to 10² (100). So, this set does not form a right triangle.
b) 36-48-60
Applying the Pythagorean theorem: 36² + 48² = 1296 + 2304 = 3600, which is equal to 60² (3600). So, this set does form a right triangle.
c) 6-10-14
Applying the Pythagorean theorem: 6² + 10² = 36 + 100 = 136, which is not equal to 14² (196). So, this set does not form a right triangle.
d) 14-48-50
Applying the Pythagorean theorem: 14² + 48² = 196 + 2304 = 2500, which is equal to 50² (2500). So, this set does form a right triangle.
In conclusion, the sets of side lengths that could form a right triangle are 36-48-60 (option b) and 14-48-50 (option d).
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An investor purchases 500 shares of Exxon-mobil stock at $98. 93 per share. His broker charges 2% of the cost of the stock. What is the cost of the stock?
The cost of the stock, including the broker's fee, is $50,454.30.
How to find the total cost of stock?The cost of the stock can be found by multiplying the number of shares purchased by the price per share. In this case, the investor purchased 500 shares of Exxon-mobil stock at $98.93 per share, so the cost of the stock can be calculated as follows:
Cost of stock = Number of shares × Price per share
Cost of stock = 500 × $98.93
Cost of stock = $49,465
However, the broker charges 2% of the cost of the stock, which is an additional fee that needs to be added to the total cost. To find the broker's fee, we can simply multiply the cost of the stock by 2%:
Broker's fee = 2% × Cost of stock
Broker's fee = 2% × $49,465
Broker's fee = $989.30
Therefore, the total cost of the stock, including the broker's fee, is:
Total cost of stock = Cost of stock + Broker's fee
Total cost of stock = $49,465 + $989.30
Total cost of stock = $50,454.30
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suppose you are playing poker with a non-standard deck of cards. the deck has 5 suits, each of which contains 12 values (so the deck has 60 cards total). how many 6-card hands are there, where you have at least one card from each suit?
The number of 6-card hands in which at least one card from each suit is equal to 8,184,220.
Total number of 6-card hands that can be formed from a deck of 60 cards is,
Using combination formula,
C(60, 6) = 50,063,860
Now, subtract the number of 6-card hands that do not contain at least one card from each suit.
There are 5 ways to choose the suit that will be missing from the hand.
Once this suit is chosen, there are 48 cards remaining in the other suits.
Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suit is,
C(48, 6) = 12,271,512
Overcounted the number of hands that are missing more than one suit.
There are C(5, 2) ways to choose 2 suits that will be missing from the hand.
Once these suits are chosen, there are 36 cards remaining in the other 3 suits.
Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suits is,
C(36, 6) = 1,947,792
We cannot have a 6-card hand that is missing more than 2 suits.
3 suits with no cards in the hand, which is not allowed.
Number of 6-card hands that have at least one card from each suit is,
C(60, 6) - 5×C(48, 6) + C(5, 2)×C(36, 6)
=50,063,860 - 5× 12,271,512 + 10 × 1,947,792
= 50,063,860 -61,357,560 + 19,477,920
= 8,184,220
Therefore, there are 8,184,220 of 6-card hands that have at least one card from each suit.
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An octagonal pyramid has a height of 12 and a side length of 4.14. find the surface area of the pyramid.
please provide steps so i can understand how it works
The surface area of octagonal pyramid with height of 12 and a side length of 4.24 is 281.477 unit²
Height of octagonal pyramid = 12
Side length of octagonal pyramid = 4.14
The surface area of octagonal pyramid is
SA = 2s²( 1 + √2) + 4sh
Here, s is side length of the octagonal pyramid = 4.14
h is height of the octagonal pyramid = 12
putting the values in the equation we get
SA = 2 × 4.14 ( 1 + √2 ) + 4 × 4.14 × 12
SA = 82.757 + 198.72
SA = 281.477
The surface area of octagonal pyramid is 281.477
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6 Which graph best represents a quadratic function with a range of all
real numbers greater than or equal to 3?
F
G
H
H
P
J
The fourth graph best represents a quadratic function with a range of all real numbers greater than or equal to 3
The graph that best represents a quadratic function with a range of all real numbers greater than or equal to 3 is a graph that opens upward and has a vertex at the point (h, k), where k is the minimum value of the function.
Since the range is all real numbers greater than or equal to 3, the minimum value occurs at or above 3.
Therefore, the vertex of the quadratic function lies on or above the horizontal line y = 3.
Hence, the fourth graph best represents a quadratic function with a range of all real numbers greater than or equal to 3
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Which statement is true considering a significance level of 5%?
A. The result is statistically significant, which implies that wearing a watch does not help people manage their time better.
B. The result is not statistically significant, which implies that this result could be due to random chance.
C. The result is statistically significant, which implies that wearing a watch helps people manage their time better.
D. The result is not statistically significant, which implies that wearing a watch does not help people manage their time better.
Given the scenerio in the picture about corn, the statement that is true looking at a significance level of 5% is The result is not statistically significant which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.
What is the does the 5% significance level mean in the context provided?Looking at the statement "The result is not statistically significant,"this means that the p-value (probability value) of the test was greater than 0.05. It could have 0.15 oe 0.2.
When a test is greater than 0.05 or 5 % significance level, it shows that the what is happening to the corn (increase in growth rate) could have been as a result of chance alone.
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Help me please first correct answer get branliest and please no essay
Answer:
6cm
Step-by-step explanation:
all the sides of a square are identical so we can assume the missing side as x
since all four sides are the same and the perimeter is the sum of all sides, we can write
x+x+x+x=24cm
4x=24cm
x=24/4
x=6cm
You deposit $2000 earned at a summer job in an account that pays 4. 2% simple interest. What is the balance in the account in 3 years? Estimate to the nearest whole number
A deposit of $2000 earning 4.2% simple interest for 3 years will have a balance of $2252. The estimated balance rounded to the nearest whole number is $2252.
To calculate the balance in the account after 3 years, we can use the formula
balance = principal x (1 + interest rate x time)
Plugging in the values, we get
balance = 2000 x (1 + 0.042 x 3)
balance = 2000 x (1 + 0.126)
balance = 2000 x 1.126
balance = 2252
Therefore, the balance in the account after 3 years is $2252.
As for the estimate, since the interest is simple, we can approximate it by multiplying the interest rate by the number of years and adding it to the principal. So, the estimate would be
estimate = principal x (1 + interest rate x time)
estimate = 2000 x (1 + 0.042 x 3)
estimate = 2000 x (1 + 0.126)
estimate = 2000 x 1.126
estimate = 2252
Rounding to the nearest whole number, the estimate is also $2252.
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-12+3(4-15)-40+10 plizz
Answer:
-12+3(-11)-40-10
Step-by-step explanation:
Answer:
Step-by-step explanation:
-12+12-45-40+1
0-85+1
-84
PLEASE HELP!! WILL GIVE BRAINLIEST!!! FIRST ANSWER GETS IT!!
The graph of f(x) and table for g(x)= f(kx) are given.
A coordinate plane with a quadratic function labeled f of x that passes through the points negative 2 comma 4 and negative 1 comma one and vertex 0 comma 0 and 1 comma 1 and 2 comma 4
x g(x)
−2 64
−1 16
0 0
1 16
2 64
What is the value of k?
k = -4
k = 4
k = -1/4
Answer:
k = 1
Step-by-step explanation:
We can use the table for g(x) = f(kx) to find the value of k.
Notice that when x = -2, we have g(-2) = f(k(-2)) = f(-2k) = 64. Similarly, when x = 2, we have g(2) = f(k(2)) = f(2k) = 64.
Using the fact that f(x) is a quadratic function, we can see that its axis of symmetry passes through the vertex at (0, 0), which means that the x-coordinate of the vertex is 0. This tells us that the coefficient of the x term in f(x) is 0, so the function can be written as f(x) = ax^2 + bx + c, where a is not equal to 0.
Using the points (−2,4), (−1,1), (0,0), (1,1), and (2,4), we can write a system of equations to solve for a, b, and c:
a(-2)^2 + b(-2) + c = 4
a(-1)^2 + b(-1) + c = 1
a(0)^2 + b(0) + c = 0
a(1)^2 + b(1) + c = 1
a(2)^2 + b(2) + c = 4
Simplifying and rearranging, we get:
4a - 2b + c = 4
a - b + c = 1
c = 0
a + b + c = 1
4a + 2b + c = 4
Substituting c = 0 into the system, we get:
4a - 2b = 4
a - b = 1
a + b = 1
4a + 2b = 4
Solving this system of equations, we get a = 1, b = -1, and c = 0.
Substituting these values into g(x) = f(kx), we get:
g(x) = f(kx) = x^2 - x
Substituting the values from the table into this equation, we get:
g(-2) = 4 = (-2)^2 - (-2) = 4k
g(2) = 4 = (2)^2 - (2) = 4k
Solving for k, we get k = 1 or k = -1/4.
However, we need to check which value of k satisfies all the points in between -2 and 2, so we can check g(-1) = 1 = (-1)^2 - (-1) = k, and g(1) = 1 = (1)^2 - (1) = k.
Thus, the value of k that satisfies all the points is k = 1, and therefore the answer is:
k = 1
We can use the information given to find the value of k.
Since the vertex of the quadratic function f(x) is at (0,0), we know that the equation for f(x) is in the form of f(x) = ax^2 for some constant a.
Using the point (-2, 4) on the graph of f(x), we can set up the equation 4 = 4a, which gives us a = 1.
So, the equation for f(x) is f(x) = x^2.
Now, we can use the table for g(x) = f(kx) to find the value of k.
When x = -2, we have g(-2) = f(k(-2)) = f(-2k) = 4k^2.
Similarly, when x = 2, we have g(2) = f(k(2)) = f(2k) = 4k^2.
We also know that g(0) = f(k(0)) = f(0) = 0, and g(-1) = f(k(-1)) = f(-k) = k^2 and g(1) = f(k(1)) = f(k) = k^2.
Using the values from the table, we can set up the following system of equations:
4k^2 = 64
k^2 = 16
0 = 0
k^2 = 16
The only solution that works for all of these equations is k = 4 or k = -4.
Therefore, the value of k is either k = 4 or k = -4.
ruby can assemble 2 22 gift baskets by herself in 7 77 minutes. emma can assemble 4 44 gift baskets by herself in 15 1515 minutes. ruby begins assembling gift baskets at 1 : 00 p.m. 1:00p.m.1, colon, 00, start text, p, point, m, point, end text, and emma begins assembling gift baskets at 1 : 15 p.m. 1:15p.m.1, colon, 15, start text, p, point, m, point, end text if they continue to work at the above rates, at what time will they finish the 5 4 th 54 th 54, start superscript, start text, t, h, end text, end superscript basket?
Ruby and Emma can assemble one gift basket in 0.1818 minutes, together. They will finish the 54th basket at 7:27 PM.
To solve the problem, we first need to find how many gift baskets Ruby and Emma can assemble in one minute.
Ruby can assemble 2/22 = 1/11 gift basket in one minute.
Emma can assemble 4/44 = 1/11 gift basket in one minute.
Together, they can assemble 1/11 + 1/11 = 2/11 = 0.1818 (rounded to four decimal places) gift baskets in one minute.
To assemble the 54th gift basket, they need to assemble 53 gift baskets before that.
53 gift baskets / 0.1818 gift baskets per minute = 291.8181 minutes
Since they start at 1:00 p.m. and Emma starts 15 minutes later, they will finish 291.8181 minutes after 1:15 p.m., which is approximately 7:27 p.m.
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what is the answer??
The equation that could represent each of the graphed polynomial function include the following:
First graph: y = x(x + 2)(x - 3)
Second graph: y = x⁴ - 5x² + 4
What is a polynomial graph?In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.
Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.
By critically observing the graph of the polynomial function shown in the image attached above, we can logically deduce that the first graph has a zero of multiplicity 1 at x = 2 and zero of multiplicity 1 at x = -3.
Similarly, we can logically deduce that the second graph has a zero of multiplicity 2 at x = 2 and zero of multiplicity 2 at x = -2.
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how to find vertex form when you have the parabola
Answer:
Step-by-step explanation: The vertex is the point at the bottom if the parabola opens up and at the top if it opens at the bottom.
PLS HELP! LAST QUESTION!
I WILL MAKE U BRAINLIST AND I NEED THIS!
PLS USE A DESMOS CALCULATOR AND SHOW ALL STEPS! I NEED IT.
Answer:
8.19 units.
Step-by-step explanation:
I didn't use desmos, i completed the question with all steps and working and didn't require desmos. Hope this Helps. Question was solved using trignometric ratios.
Please help I need it ASAP
Answer:
BC= 47.424
I believe that’s correct, but if you really need an answer just get a triangle calculator
Help right now Asapppppp
The angles that belongs to the right angles category are:
∠KER∠AREWhat makes a right angle in a circle?KER must be a right angle because it is an angle formed between a tangent (KE) and a radius (AK) at the point of tangency (K). By the Tangent-Secant Theorem, it follows that the measure of the intercepted arc KR is equal to the measure of the angle ∠KER plus 90 degrees. Since KR is a diameter (and therefore a semicircle), its measure is 180 degrees. Therefore, ∠KER + 90 = 180, which implies that ∠KER = 90 degrees.
∠ARE must be a right angle because it is an inscribed angle that intercepts the diameter KR. By the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc. Since KR is a diameter, the intercepted arc is the entire circle, which has a measure of 360 degrees. Therefore, ∠ARE = 360/2 = 180 degrees. Moreover, a diameter and a chord that contains the diameter must form a right angle at the point where they meet (in this case, point A). Hence, ∠ARE is a right angle.
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Image transcribed:
The figure below shows a circle with center A, diameter KR, and secants RE and RI. Which of the angles must be right angles? Select all that apply.
Q
E
R
K
D
∠KIR
∠ARD
∠KER
∠ARE
∠ERI
Daniel just graduated college and found a job that pays him $42,000 a year, and the company will give him a pay increase of 6. 5% every year. How much will Daniel earn in 4 years?
With the given pay increase Daniel will earn a total of $185,141.90 in 4 years .
To find out how much Daniel will earn in 4 years with a starting salary of $42,000 and a 6.5% pay increase every year, follow these steps:
1. Calculate the annual salary for each year by applying the percentage increase.
2. Sum up the salaries for all 4 years.
Step 1: Calculate the annual salary for each year
Year 1: $42,000
Year 2: $42,000 * (1 + 6.5%) = $42,000 * 1.065 = $44,730
Year 3: $44,730 * (1 + 6.5%) = $44,730 * 1.065 = $47,656.95
Year 4: $47,656.95 * (1 + 6.5%) = $47,656.95 * 1.065 = $50,754.95
Step 2: Sum up the salaries for all 4 years
Total earnings = $42,000 + $44,730 + $47,656.95 + $50,754.95 = $185,141.90
Daniel will earn a total of $185,141.90 in 4 years with the given pay increase.
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What’s the answer? I need help please
Answer: 10/12
Step-by-step explanation:
since they give you adjacent to angle m and hypotenuse use
cos x = opp/hyp
cos M = 10/12
For how many different integers $k$ are there rational solutions to the quadratic equation
[tex]\[x^2 + kx + 4k = 0?\][/tex]
For k = 0 and k = 16, there are rational solutions to the quadratic equation [tex]x^{2} + kx + 4k = 0[/tex]
We are given a quadratic equation [tex]x^{2} + kx + 4k = 0[/tex]
An algebraic equation in x with a degree of 2 is known as a quadratic equation. It is written in the format [tex]a[/tex][tex]x^{2}[/tex] [tex]+ bx + c[/tex] = 0. To find out whether there exists two solutions, one solution, or no solution for a quadratic equation, we use the discriminant of the quadratic equation.
We will find the solutions to this quadratic equation with the help of discriminant formula
As we know from the equation that b = k, a = 1, and c = 4k.
[tex]b^2 - 4ac = 0[/tex]
[tex]k^2 - 4(4k) = 0[/tex]
[tex]k^2 - 16k = 0[/tex]
k (k-16) = 0
k = 0 or k - 16 = 0
k = 0 or k = 16
So, for k = 0 or k = 16 the equation [tex]x^{2} + kx + 4k = 0[/tex] has only one solution.
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The base and all three faces of a triangle pyramid are equilateral triangles with side lengths of 3ft. the height of each triangle is 2.6ft. what are the lateral area and the total surface area of the triangular pyramid?
The lateral area of the triangular pyramid is 11.7 sq ft and the total surface area is 15.6 sq ft.
To find the lateral area and total surface area of the triangular pyramid with base and faces as equilateral triangles, we can follow these steps:
1: Find the area of one equilateral triangle.
To find the area of an equilateral triangle with side length 3 ft and height 2.6 ft, we can use the formula:
Area = (1/2) × base × height
Area = (1/2) × 3 × 2.6 = 3.9 sq ft
2: Calculate the lateral area.
Since the pyramid has three equilateral triangles as faces, we can multiply the area of one triangle by 3 to find the lateral area:
Lateral Area = 3 × 3.9 = 11.7 sq ft
3: Calculate the total surface area.
The total surface area includes both the lateral area and the base area. Since the base is also an equilateral triangle with the same dimensions, we can simply add the area of the base to the lateral area to find the total surface area:
Total Surface Area = Lateral Area + Base Area = 11.7 + 3.9 = 15.6 sq ft
In conclusion, the lateral area is 11.7 sq ft and the total surface area is 15.6 sq ft.
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