Step-by-step explanation:
h = -490t² + 1470t
When h = 980:
980 = -490t² + 1470t
Simplifying:
0 = -490t² + 1470t - 980
0 = t² - 3t + 2
Factoring:
0 = (t - 1) (t - 2)
t = 1, 2
There are two solutions because the rocket first reaches the height of 980 cm as it's going up at 1 second, then it reaches that height again as it's coming down at 2 seconds.
The rocket reaches a height of 980 cm at 1 second and 2 seconds. The equation is factored to find these solutions, symbolizing the ascent and descent of the rocket.
To find when the height of the rocket is 980 cm, we set the height formula equal to 980 cm:
h = -490t² + 1470t = 980.
We then solve the equation by factoring:
-490t² + 1470t - 980 = 0
Divide by -10:
t²- 3t + 2 = 0
Factor:
(t - 1)(t - 2) = 0
Setting each factor to zero gives us the solutions:
t = 1 second
t = 2 seconds
There are two solutions because the rocket reaches 980 cm twice: once on its way up and once on its way down. This is demonstrated by the positive and negative components of the parabolic trajectory represented by the quadratic equation.
what is the area of isosceles trapazoid MNOP?
Answer:
340 units squared
Step-by-step explanation:
in the trapezoid, there is a square and an isosceles triangle
find the area of the square: 17×(23-6) =17×17 =289
find area of the triangle: 17×6 /2 =51
add the two together: 289 +51 =340
Answer:
Line MN = 23 -3 -3 = 17
Middle Area = 17 * 17 = 289
Area of BOTH Ends = 17 * 3 = 51
TOTAL Area = 289 + 51 = 340
Step-by-step explanation:
Please help me solve this problem the quickest and shortest way.
Thank you so much!
Answer options are :
-5
-2
1
7
Answer:
-5
Step-by-step explanation:
Sometimes f(x) can be a little confusing. You never quite know where which number goes where.
Start with the first part
f(6) means that wherever you see an x on the right you put a 6.
So do this.
f(6) = (3/2)(6) +b =
equals what
(3/2) * 6 + b = 7 That's because f(6) = 7 Now just solve it
(18)/2 + b = 7
9 + b = 7
9 - 9 + b = 7 - 9
b = - 2
================
The entire equation is
y = (3/2)x - 2
===============
What happens when you are given
f(-2)
The - 2 means that wherever you see an x, let it be - 2
f(-2) = (3/2)(-2) - 2
f(-2) = -3 - 2
f(-2) = - 5
===============
You could think of f(x) being y. You will not go wrong doing that. y = f(x)
In the figure below, point H is the incenter of DEF, and m DHF=130. What is the measure of DEF?
Answer:
The measure of angle DEF is ∠DEF=80°
Step-by-step explanation:
we know that
The incenter is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle
The triangle DEF is an isosceles triangle
∠EDF=∠EFD
so
In the triangle DHF
∠HDF=∠HFD
∠HDF+∠DHF+∠HFD=180°
substitute the value
2∠HDF+130°=180°
∠HDF=25°
therefore
∠EDF=2*25°=50°
In the triangle DEF
∠EDF+∠DEF+∠EFD=180°
∠EDF=∠EFD=50°
substitute
50°+∠DEF+50°=180°
∠DEF=80°
Answer:
The answer would be 80?
That is correct!
Step-by-step explanation:
Once Im done with the question Ill come back and make sure its correct!
A car loses its 15% of its current value every year. What is the value of a car that was bought for $20,000 in two years?
Answer:
$14,450
Step-by-step explanation:
y=20,000(1-0.15)^x
Hope This Helps :D
The critical value for a two-tailed test of h0: β1 = 0 at α = .05 in a simple regression with 22 observations is:
The critical value for a two-tailed test of h0: β1 = 0 at α = .05 in a simple regression with 22 observations is 4.35.
What is a critical value?A critical value can be calculated for different types of hypothesis tests. The critical value of a particular test can be interpreted from the distribution of the test statistic and the significance level.
Test of significance of linear regression between x and y.
Null hypothesis: There is no significant linear relationship between x and y.
Alternative hypothesis: There is a significant linear relationship between x and y.
Test statistic:
F0 = Mean square due to regression/ mean squared error ~ F(1, n-2)
Critical value: [tex]F_{\alpha}(1, n-2) = F_{0.05}(1,20) = 4.35[/tex]
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If a yogurt is 150 calories per 2/3 cup, how many calories is a full cup?
Thanks for the help!
Answer:
The answer is 225
Step-by-step explanation:
Setup a ratio saying 150:2/3
Next divide 2/3 by 1 to see how many times it can go into it which is 1 1/2
After that multiply 1 1/2 by 150
Answer is 225.
If a yogurt is 150 calories per 2/3 cup, how many calories is a full cup?
If we know that 150 calories are in 2/3 of a cup of yogurt, we can find out how many calories are in a full cup. If there are 150 calories in 2/3 of a cup, let’s first find out how many calories are in 1/3 of a cup by dividing 150 by 2.
150 ÷ 2 = 75 calories per 1/3 of a cup.
Now, we can multiply 75 by 3 to find out how many calories are in a full cup of yogurt.
75 x 3 = 225 calories per 1 cup of yogurt.
Therefore, there are 225 calories in 1 full cup of yogurt.
Develop a 95% confidence interval for the expected value of y when x = 8. estimate the standard deviation of an individual value of y when x = 8.
Final answer:
To create a 95% confidence interval for y when x=8, we need the sample mean and the standard deviation to calculate the Error Bound for the Mean (EBM). The CI is the sample mean ± EBM. Estimating the standard deviation of y involves using the sample standard deviation as an estimate or the known population standard deviation.
Explanation:
To develop a 95% confidence interval for the expected value of y when x = 8, we first need to know the sample mean (μ) and the standard deviation of the population or an estimate from the sample (known as σ or s, respectively). When we have these statistics, we can use the normal distribution to calculate the error bound for the mean (EBM), which is the margin of error. The confidence interval (CI) is then constructed as (sample mean - EBM, sample mean + EBM).
To estimate the standard deviation of an individual value of y when x = 8, which is essentially the standard deviation of the population (σ), you need to use the sample standard deviation (s) as an estimate unless the population standard deviation is known.
Remember, the EBM for the confidence interval depends on the chosen confidence level, the standard deviation, and the size of the sample. With a higher confidence level, you get a wider interval. In practice, the confidence interval gives us a range where we expect the true population parameter lies with a certain level of confidence, acknowledging there's still a small chance the true value lies outside of this range.
Are data at the nominal level of measurement quantitative or qualitative?
a. both qualitative and quantitative
b. qualitative
c. quantitative
d. neither qualitative nor quantitative
Answer:
B. qualitative
Step-by-step explanation:
The table of values represents the function g(x) and the graph shows the function f(x).
Which statements are true?
Select EACH correct answer.
A. g(x) has fewer x-intercepts than f(x).
B. f(x) and g(x) have a common x-intercept.
C. The maximum value of g(x) is greater than the maximum value of f(x) .
D. f(x) has a greater y-intercept than g(x).
Answer:
B, C
Step-by-step explanation:
x-intercepts are when y=0. f(x) has two x-intercepts at (1, 0) and (5, 0). g(x) also has two x-intercepts; (-3, 0) and (5, 0). So the first one is false, and the second one is true.
The maximum value of f(x) is 2. The maximum value of g(x) is 4. So the third one is true.
The y-intercept is the value of y when x=0. So the y-intercept of f(x) is -1, and the y-intercept of g(x) is 3. So the fourth one is false.
The statement which is true are f(x) and g(x) have a common x-intercept (B) and the maximum value of g(x) is greater than the maximum value of f(x) (C).
What is x-intercept?The x-intercept is the point on the coordinate at which a line, curve or plane intersect with the x-axis. The value of y is equal to zero at x-intercept.
The function f(x) shown in the graph has two x intercept (5,0) and (1,0). The x intercept of g(x) are (-3, 0) and (5, 0).
Similarly, the y-intercept is the point on the coordinate at which a line, curve or plane intersect with the y-axis. The value of x is equal to zero at y-intercept.
The function f(x) shown in the graph has one y intercept (0,-1). The y intercept of g(x) is (0, 3). Here the maximum value of f(x) is 2 while g(x) is 4.
Thus, the statement which is true are f(x) and g(x) have a common x-intercept (B) and the maximum value of g(x) is greater than the maximum value of f(x) (C).
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Someone please help??
Answer:
x-axis
Step-by-step explanation:
The asymptote is a straight line that the curve gets closer and closer to but never touches it.
The given exponential function is [tex]f(x)=3^x[/tex].
The given graph has a horizontal asymptote,
The equation of this horizontal asymptote is y=0.
This is also refers to as the x-axis.
Therefore the asymptote is the x-axis.
The lines shown below are perpendicular. If the green line has a slope of -1/4 what is the slope of the red line
Answer:
4
Step-by-step explanation:
Since perpendicular lines have negative reciprocals, we have to find out the negative reciprocal of -1/4.
To get the negative reciprocal of -1/4, we have to switch the numbers in the numerator and denominator.
That becomes 4/-1, and then we have to remove the negative, which makes the final answer 4/1.
4/1 = 4
Therefore, the negative reciprocal of -1/4 is 4.
Hope this Helps!
The slope of the red perpendicular line is 4.
What is slope of a perpendicular line ?Slope of a straight line is defined as the inclination of the line with respect to the coordinate axis. The slope is also measure of the tangent of angle with which the line is inclined to the axis.
If the slope of a given straight line is m then the slope of its corresponding perpendicular line will be negative reciprocal of the given slope which is -(1/m).
How to find the slope of the given curve ?It is said that the lines shown below are perpendicular. Also it is mentioned that the green line has a slope of -1/4 .
As the red line is perpendicular to the green line, thus its slope is negative reciprocal of the given slope of green line.
Slope = -(-4) = 4 .
Therefore, the slope of the red perpendicular line is 4.
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Algebra Help Please
Moira buys a new rectangular rug that is 3 inches shorter than her old rectangular rug. The area of the new rug is (x2 + 2x - 15)in2.
Describe the width of Moira’s new rug in terms of the length of her old rug.
Moira’s new rug is (blank) inches (blank) than the length of her old rug.
Choices 7, 5, 0, 15 | Narrower, Wider
Answer:
5 inches wider
Step-by-step explanation:
The given area expression factors as ...
x² +2x -15 = (x -3)(x +5)
Apparently, we're to assume that these factors represent length times width, and the factor x-3 corresponds to a length that is 3 inches shorter than the old length (x). Then the width is (x+5), which is 5 inches wider than the old length.
Answer:
5 in wider is right
Step-by-step explanation:
It costs 9.95 for 1 ticket to the movies. If 3 people go, how much would the total price of 3 tickets be?
It would be $9.95 x 3
Suppose the integral from 2 to 8 of g of x, dx equals 12, and the integral from 6 to 8 of g of x, dx equals negative 3, find the value of 2 times the integral from 2 to 6 of g of x, dx .
7.5
15
18
30
[tex]\displaystyle\int_2^8g(x)\,\mathrm dx=12[/tex]
[tex]\displaystyle\int_6^8g(x)\,\mathrm dx=-3[/tex]
Use the fact that integrals are additive on their intervals. Mathematically, if [tex]c\in[a,b][/tex], then
[tex]\displaystyle\int_a^bg(x)\,\mathrm dx=\int_a^cg(x)\,\mathrm dx+\int_c^bg(x)\,\mathrm dx[/tex]
So we have
[tex]\displaystyle2\int_2^6g(x)\,\mathrm dx=2\left(\int_2^8g(x)\,\mathrm dx-\int_6^8g(x)\,\mathrm dx\right)=2(12-(-3))=30[/tex]
The value of 2 times the integral from 2 to 6 of g(x) dx is 30.
The integral from 2 to 8 of g(x) dx = 12
The integral from 6 to 8 of g(x) dx = -3
We want to find:
2 times the integral from 2 to 6 of g(x) dx.
First, we'll use the properties of integrals:
The integral from 2 to 8 of g(x) dx = The integral from 2 to 6 of g(x) dx + The integral from 6 to 8 of g(x) dx.
We know that the integral from 2 to 8 of g(x) dx is 12, and the integral from 6 to 8 of g(x) dx is -3:
12 = The integral from 2 to 6 of g(x) dx - 3.
Next, we'll isolate the integral we're interested in:
The integral from 2 to 6 of g(x) dx = 12 + 3
= 15.
Finally, we'll multiply this result by 2:
2 * 15 = 30.
So, 2 times the integral from 2 to 6 of g(x) dx is 30. The correct answer is 30.
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Where should you plot the ordered pair (3,4) in relation to the origin?
A. 4 spaces to the right and 3 spaces up
B. 3 spaces to the right and 4 spaces up
C. 3 spaces to the right, 4 times
D. 3 spaces up and 4 spaces to the right
Answer:
B. 3 spaces to the right and 4 spaces up
Step-by-step explanation:
With both coordinates positive, we know that this point (3, 4) is in Quadrant I.
B. 3 spaces to the right and 4 spaces up
is the correct choice.
Determine whether the situation involves a permutation or a combination, and how many possibilities are there are?
A team of six students is chosen from a class of 36.
Answer:
This situation involves a combination as the team will be the same no matter what order the students are chosen in.
There are 1402410240 possibilities for the six student team
Step-by-step explanation:
Each time a student is chosen, they cannot be chosen again, so the number of available students decreases each time that one is chosen.
[tex]36*35*34*33*32*31=1402410240[/tex]
What is the angle of depression from the top of a 500-foot cable that runs from a tower to a point 350 feet away?
Answer:
45.6 degrees
Step-by-step explanation:
First of all this needs to be recognized as a right triangle problem. The angle of depression is the angle outside the triangle that is complementary to the vertex angle (the angle at the top). Geometrically, this angle is congruent to the base angle inside the triangle (NOT the right angle). So we need to find the measure of the base angle to find the measure of the angle of depression since they are the same. A 500 foot cable describes the length of the cable, which serves as our hypotenuse. The 350 is the base length of the triangle. What we have then is a reference angle (x), the hypotenuse (500) and the side adjacent to the angle (350). The ratio that relates the angle to the hypotenuse and the adjacent side is the cosine. Setting up to find our angle gives us this equation:
[tex]cos\theta=\frac{350}{500}[/tex]
You find missing angles on your calculator by hitting the 2nd button and then the trig identity you want. We want cosine, so hit 2nd then cos and you'll get cos^-1( on your screen. After the parenthesis, enter your 500/350 and hit enter. This will give you your angle measure of 45.6. Oh yeah!!! And make sure your calculator is in degree mode for this one!
The angle of depression from the top of a 500-foot cable that runs from a tower to a point 350 feet away is 45.6 degrees
The situation forms a right angle triangle.
The length of the cable is the hypotenuse of the triangle formed.
The distance from the tower is the adjacent side of the right angle triangle.
Therefore, using trigonometric ratio,
let
∅ = angle of depression
cos ∅ = adjacent / hypotenuse
cos ∅ = 350 / 500
∅ = cos⁻¹ 0.7
∅ = 45.5729959992
∅ = 45.6°
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A 52 year-old father has a son and a daughter. The son is twice as old as the daughter. In 4 years the sum of all their ages will be 100. How old are the two siblings now?
Answer:
24 and 12
Step-by-step explanation:
s=2d
s+d+52+4(3)=100
Answer:
son is 24 years old and daughter is 12 years old.
Step-by-step explanation:
Let the age of son = x years and age of the daughter = y years.
Now for equations
" The son is twice as old as the daughter "
x = 2y -------(1)
"After 4 years sum of all their ages will be 100."
56 + (x+4) + (y+4) = 100
x + y + 64 = 100
x + y = 100 - 64 = 36 ----------(2)
Now we substitute the values of x from equation (1) to (2)
2y + y = 36
3y = 36
y = 12
Now from equation (1)
x = 2 × 12
x = 24 years
Therefore, son is 24 years old and daughter is 12 years old.
Saul threw a party for 7 people. Each person drank 2/3 cup of lemonade. Together, how many cups of lemonade did they drink?
Find the product of 7 and 2/3. When multiplying fractions, first multiply the numerators, and then multiply the denominators.
7/1*2/3
7*2=14
1*3=3
14/3 cups of lemonade
Hope this helps!!
To find the total amount of lemonade consumed by 7 people drinking 2/3 cup each, calculate the amount per person, then multiply by the number of people for the total to get 14/3 or 4 2/3 as the total lemonade consumed.
Saul threw a party for 7 people. Each person drank 2/3 cup of lemonade.
To find the total amount of lemonade consumed:
Calculate the amount of lemonade each person drank: 2/3 cup/person.
Find the total amount for all 7 people: 7 people x 2/3 cup/person = 14/3 cups.
Therefore, the total amount of lemonade consumed is 14/3 or 4 2/3 cups
Solve the system of equations. y = -6x + 9 y = -3x + 3 a. ( 3, -2) c. ( 2, -3) b. ( -2, -3) d. No solution
For this case we have the following system of equations:
[tex]y = -6x + 9\\y = -3x + 3[/tex]
Equating the equations we have:
[tex]-6x + 9 = -3x + 3[/tex]
Adding 3x to both sides of the equation:
[tex]-6x + 3x + 9 = 3\\-3x + 9 = 3[/tex]
Subtracting 9 from both sides of the equation:
[tex]-3x = 3-9\\-3x = -6[/tex]
Dividing between -3 on both sides of the equation:
[tex]x = \frac {-6} {- 3}\\x = 2[/tex]
We look for the value of "and":
[tex]y = -3 (2) +3\\y = -6 + 3\\y = -3[/tex]
Thus, the solution of the system is (2, -3)
Answer:
(2, -3)
5.14x+1.76+0.9x−x=32
Answer:
x=6
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Don't understand what I have to do in order to find out what I have to plot.
These are the values in Kamil’s data set.
(1, 22), (3, 18), (5,14), (6, 13), (8,5)
Kamil determined the equation of a linear regression line and determined that these are the predicted values, to the nearest integer.
(1, 23), (3, 18), (5,14), (6, 11), (8,7)
(1, 22), (3, 18), (5,14), (6, 13), (8,5)
(1, 23), (3, 18), (5,14), (6, 11), (8,7)
you plot there on a graph. do you know how to place on a graph?
if not, the line going up and down is the x-axis and the line going side to side is the y-axis,
for example (4,7) it would be (x,y)
i really hope this helps :)
Suppose a triangle has two sides of length 33 and 37 and that the angle between these two sides is 120 what is the length of the third side of the triangle
Answer:
60.65
Step-by-step explanation:
The Law of Cosines can help you figure this out. Call the given sides "a" and "b" and the given angle "C". Then the third side, "c" will satisfy the relation ...
c² = a² + b² -2ab·cos(C)
= 33² +37² -2·33·37·cos(120°) = 3679
c = √3679 ≈ 60.65476 ≈ 60.65
The length of the third side is about 60.65 units.
Matt paid $55 at a restaurant. That amount included a 10% tip. What was the check amount before the tip? A. $40 B. $43 C. $45 D. $50 E. $52
Answer:
D $50
Step-by-step explanation:
50x.10=5
5+50=55
For this case we propose a rule of three:
$ 50 -------------> 100%
x -----------------> 10%
Where the variable "x" represents the amount of the tip.
[tex]x = \frac {10 * 50} {100}\\x = 5[/tex]
So, the tip was 5 dollars.
If we subtract that amount to what Matt paid at the restaurant we will have the amount of the check before the tip.[tex]55-5 = 50[/tex]
ANswer:
Option D
Please help me out please
Answer is A) Acute due to the lengths
Answer:
A
Step-by-step explanation:
Sandra's printer can print 5 pages in 2/3 of a minute (40s). Show different ways to determine how many pages her printer can print in 1 minute.
Answer:
[tex]7\frac{1}{2}[/tex] pages
Step-by-step explanation:
Method 1: Working with minutes
Let us assume that, Sandra's printer can print x pages in 1 minute.
Then we can write the equation:
[tex]x:1=5:\frac{2}{3}[/tex]
We can rewrite the ratios as a fraction.
[tex]\frac{x}{1}=\frac{5}{\frac{2}{3}}[/tex]
This simplifies to
[tex]x=\frac{15}{2}[/tex]
Or
[tex]x=7\frac{1}{2}[/tex]
Method 2: Working with seconds
If 5 pages corresponds to 40 seconds, then we can write: [tex]5:40[/tex]
Let y corresponds to 60 seconds (1 minutes).
Then:
[tex]y:60=5:40[/tex]
We rewrite as fraction:
[tex]\frac{y}{60}=\frac{5}{40}[/tex]
Multiply both sides by:
[tex]y=\frac{5}{40}\times 60[/tex]
[tex]y=\frac{300}{40}[/tex]
[tex]y=\frac{15}{2}[/tex]
[tex]y=7\frac{1}{2}[/tex]
Evaluate the surface integral. s xy ds s is the triangular region with vertices (1, 0, 0), (0, 8, 0), (0, 0, 8)
To evaluate the surface integral [tex]\( \iint_S xy \, dS \)[/tex] over the triangular region with vertices[tex]\((1,0,0)\), \((0,8,0)\), and \((0,0,8)\):[/tex]
1. **Equation of the plane**:
8x + y + z = 8.
2. **Parametrize the surface**:
[tex]\[ \mathbf{r}(x, y) = (x, y, 8 - 8x - y). \][/tex]
3. **Normal vector**:
[tex]\[ \mathbf{N} = (8, 8, 1), \quad |\mathbf{N}| = \sqrt{129}. \][/tex]
4. **Surface integral setup**:
[tex]\[ \iint_D xy \sqrt{129} \, dA, \quad \text{with} \quad D: \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
5. **Evaluate the integral**:
[tex]\[ \sqrt{129} \int_0^1 32x (1-x)^2 \, dx = \frac{8 \sqrt{129}}{3}. \][/tex]
Thus, the value of the surface integral [tex]\( \iint_S xy \, dS \) is \( \frac{8 \sqrt{129}}{3} \).[/tex]
To evaluate the surface integral [tex]\( \iint_S xy , dS )[/tex], where ( S ) is the triangular region with vertices (1, 0, 0) , (0, 8, 0) , and (0, 0, 8) , we can follow these steps:
1. Determine the equation of the plane containing the triangle:
The vertices of the triangle are (1, 0, 0), (0, 8, 0), and (0, 0, 8). The equation of the plane can be found using these points.
The general form of a plane equation is (Ax + By + Cz = D).
Substituting the points:
[tex]- For \((1, 0, 0)\): \(A(1) + B(0) + C(0) = D \rightarrow A = D\).[/tex]
[tex]- For \((0, 8, 0)\): \(A(0) + B(8) + C(0) = D \rightarrow 8B = D \rightarrow B = \frac{D}{8}\).[/tex]
[tex]- For \((0, 0, 8)\): \(A(0) + B(0) + C(8) = D \rightarrow 8C = D \rightarrow C = \frac{D}{8}\).[/tex]
So, (D = A), [tex]\(B = \frac{A}{8}\), and \(C = \frac{A}{8}\).[/tex]
Using (A = 8) (chosen for simplicity):
- (A = 8),
- (B = 1),
- (C = 1),
- (D = 8).
Thus, the plane equation is (8x + y + z = 8).
2. Parametrize the surface:
We can use (x) and (y) as parameters. From the plane equation, we get (z = 8 - 8x - y).
Let ( (x, y) ) be the parameters. The surface can be parametrized as:
[tex]\[ \mathbf{r}(x, y) = (x, y, 8 - 8x - y). \][/tex]
3. Find the normal vector and its magnitude:
The normal vector [tex]\( \mathbf{N} \)[/tex] can be found from the cross product of the partial derivatives of [tex]\(\mathbf{r}\)[/tex] with respect to (x) and (y).
[tex]\[ \mathbf{r}_x = \frac{\partial \mathbf{r}}{\partial x} = (1, 0, -8), \][/tex]
[tex]\[ \mathbf{r}_y = \frac{\partial \mathbf{r}}{\partial y} = (0, 1, -1). \][/tex]
[tex]\[ \mathbf{N} = \mathbf{r}_x \times \mathbf{r}_y = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & -8 \\ 0 & 1 & -1 \\ \end{vmatrix} = (8, 8, 1). \][/tex]
The magnitude of [tex]\( \mathbf{N} \)[/tex] is:
[tex]\[ |\mathbf{N}| = \sqrt{8^2 + 8^2 + 1^2} = \sqrt{64 + 64 + 1} = \sqrt{129}. \][/tex]
4. Surface integral:
The surface integral is given by:
[tex]\[ \iint_S xy \, dS = \iint_D xy |\mathbf{N}| \, dA, \][/tex]
where (D) is the projection of (S) on the (xy)-plane.
From the vertices, (D) is a triangular region with vertices (1,0), ((0,8), and (0,0).
5. Set up the integral in (xy) coordinates:
We can describe (D) as:
[tex]\[ \iint_D xy \, \sqrt{129} \, dA. \][/tex]
The bounds for (x) and (y) are:
[tex]\[ \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
6. Evaluate the integral:
[tex]\[ \sqrt{129} \int_0^1 \int_0^{8(1-x)} xy \, dy \, dx. \][/tex]
Evaluate the inner integral with respect to (y):
[tex]\[ \int_0^{8(1-x)} xy \, dy = x \left[ \frac{y^2}{2} \right]_0^{8(1-x)} = x \left( \frac{(8(1-x))^2}{2} \right) = x \left( \frac{64(1-x)^2}{2} \right) = 32x (1-x)^2. \][/tex]
So the integral becomes:
[tex]\[ \sqrt{129} \int_0^1 32x (1-x)^2 \, dx. \][/tex]
Let ( u = 1-x ). Then ( du = -dx ), and the limits change from ( x = 0 ) to ( u = 1 ), and ( x = 1 ) to ( u = 0 ):
[tex]\[ \sqrt{129} \int_1^0 32(1-u) u^2 (-du) = \sqrt{129} \int_0^1 32(1-u) u^2 \, du. \][/tex]
Expand and integrate:
[tex]\[ \sqrt{129} \int_0^1 32 (u^2 - u^3) \, du = 32 \sqrt{129} \left[ \frac{u^3}{3} - \frac{u^4}{4} \right]_0^1 = 32 \sqrt{129} \left( \frac{1}{3} - \frac{1}{4} \right). \][/tex]
Simplify:
[tex]\[ 32 \sqrt{129} \left( \frac{4}{12} - \frac{3}{12} \right) = 32 \sqrt{129} \left( \frac{1}{12} \right) = \frac{32 \sqrt{129}}{12} = \frac{8 \sqrt{129}}{3}. \][/tex]
Thus, the value of the surface integral[tex]\( \iint_S xy \, dS \) is \( \frac{8 \sqrt{129}}{3} \).[/tex]
The graph of which equation below is a horizontal line?
Answer:
2
Step-by-step explanation:
The equation of a horizontal line is y = c
where c is the value of the y- coordinates the line passes through
Thus y = 3 ← is the equation of a horizontal line
How much wrapping is needed to cover a cubed gift box that is 9 inches high? (include the bow which takes 115 sq. inches.)?
Answer:
601 square inches
Step-by-step explanation:
The wrapping is needed to cover a cubed gift box that is 9 inches high is 616 sq inches.
What is surface area? The surface area is the sum of the areas of all its faces.The areas of the base, top, and lateral surfaces i.e all sides of the object. It is measured using different area formulas and measured in square units and then adding all the areas. The surface area of a solid object is a measure of the total area that the surface of the object covers.Calculation:-
The area of gift wrapping needed is equal to the total surface area of the box.
area of a bow =115
∴ side of bow =[tex]\sqrt{115}[/tex]
= 10.72
total surface area = 115+115+10.72+10.72
=616 sq ft.
Learn more about the surface area here:- https://brainly.com/question/16519513
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Emily wants to hang a painting in a gallery. The painting and frame must have an area of 31 square feet. The painting is 5 feet wide by 6 feet long. Which quadratic equation can be used to determine the thickness of the frame, x? (5 points)
Answer:
4x² + 22x − 1 = 0
Step-by-step explanation:
Start off with the standard formula to finding area with polynomials and quadratic functions.
(2x + 5)(2x + 6)
Work out like a normal binomial.
4 + 22x + 30 - 31
(You're subtracting 31 to take out the minimum of what the frame has to be)
4 + 22x - 1 = 0
Answer:
4x² + 22x − 1 = 0