The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.
What is mean by Rectangle?
A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.
Given that;
The rectangle is bounded by the x - axis and the semicircle y = 49 - x².
Since,
The area of rectangle with sides x and y is,
Area = x × y
A = xy
Since, The equation of the semicircle is;
y = 49 - x².
Substitute the values of y in equation (i), we get;
A = x (49 - x²)
A = 49x - x³
Now, Find the derivative and equate into zero,
A' = 49 - 3x²
A' = 0
49 - 3x² = 0
49 = 3x²
x² = 49/3
x = 7/√3
x = 7/1.73
x = 4.04
Hence, y = 49 - x²
y = 49 - (4.04)²
y = 49 - 16.3
y = 32.67
Since, The area is maximum when we can multiply x by y as;
Maximum area = 4.04 x 32.67
Maximum area = 132
Hence, The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.
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I'm not sure how to do this. This is a long one that's why.
We have the following:
For the area surface:
[tex]As=2\pi rh[/tex]repacing:
r = 1.5 in
h = 7 in
[tex]\begin{gathered} As=2\cdot3.14\cdot1.5^{}\cdot7 \\ As=65.94 \end{gathered}[/tex]The answer is 65.9 in^2
For volume:
[tex]\begin{gathered} V=\pi r^2h \\ V=3.14\cdot1.5^2\cdot7 \\ V=49.455 \end{gathered}[/tex]The answer is 49.5 in^3
Describe in words where the square root of 60 minus 11 would be plotted on a number line.
Answer:
it would be on 7 since 60-11=49 and the square root of 49 is 7
NO LINKS!! Describe the domain and range (in BOTH interval and inequality notation) for each function shown part 1a
Answer:
Domain as an inequality: [tex]\boldsymbol{\text{x} < 6 \ \text{ or } \ -\infty < \text{x} < 6}[/tex]
Domain in interval notation: [tex]\boldsymbol{(-\infty, 6)}[/tex]
Range as an inequality: [tex]\boldsymbol{\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6}[/tex]
Range in interval notation: [tex]\boldsymbol{(-\infty, 6]}[/tex]
=========================================================
Explanation:
The domain is the set of allowed x inputs. For this graph, the right-most point is when x = 6. This endpoint is not part of the domain due to the open hole. The graph goes forever to the left to indicate [tex]\text{x} < 6[/tex] but I think [tex]-\infty < \text{x} < 6[/tex] is far more descriptive.
The second format directly leads to the interval notation of [tex](-\infty, 6)[/tex]
Always use parenthesis for either infinity. We use a parenthesis for the 6 to tell the reader not to include it as part of the domain.
------------------------
The range is the set of possible y outputs.
The highest y can get is y = 6
Therefore, y = 6 or y < 6
The range can be described as [tex]\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6[/tex] where the second format is better suited to lead directly to the interval notation [tex](-\infty, 6][/tex]
Use a square bracket to include the 6 as part of the range. We don't have any open holes at the peak mountain point.
Answer:
[tex]\textsf{Domain}: \quad (-\infty, 6) \quad -\infty < x < 6[/tex]
[tex]\textsf{Range}: \quad (-\infty,6] \quad -\infty < y\leq 6[/tex]
Step-by-step explanation:
The domain of a function is the set of all possible input values (x-values).
The range of a function is the set of all possible output values (y-values).
An open circle indicates the value is not included in the interval.
A closed circle indicates the value is included in the interval.
An arrow show that the function continues indefinitely in that direction.
Interval notation
( or ) : Use parentheses to indicate that the endpoint is excluded.[ or ] : Use square brackets to indicate that the endpoint is included.Inequality notation
< means "less than".> means "more than".≤ means "less than or equal to".≥ means "more than or equal to".From inspection of the given graph, the function is not continuous and so the domain is restricted.
There is an open circle at x = 6.
Therefore, the domain of the function is:
Interval notation: (-∞, 6)Inequality notation: -∞ < x < 6From inspection of the given graph, the maximum value of y is 6.
The function continues indefinitely to negative infinity.
Therefore, the range of the function is:
Interval notation: (-∞, 6]Inequality notation: -∞ < y ≤ 6Sam goes to a fast food restaurant and orders some tacos and burritos. He sees on the nutrition menu that tacos are 250 calories and burritos are 330 calories. If he ordered 4 items and consumed a total of 1080 calories, how many tacos, and how many burritos did Sam order and eat?
Let x represent the number of tacos that Sam ordered and ate.
Let y represent the number of burritos that Sam ordered and ate.
From the information given, If he ordered 4 items, it means that
x + y = 4
If tacos are 250 calories, it means that the number of calories in x tacos is 250 * x = 250x
If burritos are 330 calories, it means that the number of calories in y burritos is 330 * y = 330y
If he consumed a total of 1080 calories, it means that
250x + 330y = 1080
From the first equation, x = 4 - y
By substituting x = 4 - y into the second equation, we have
250(4 - y) + 330y = 1080
1000 - 250y + 330y = 1080
- 250y + 330y = 1080 - 1000
80y = 80
y = 80/80
y = 1
x = 4 - y = 4 - 1
x = 3
Thus, Sam ordered and ate 3 tacos and 1 burritos
Which is equal to 2 over 5? A. 2%B. 2.5%C. 20%D. 25%E. 40%
Calculating the value of 2 over 5 in percentage, we have:
[tex]\begin{gathered} \frac{2}{5}=\frac{20}{50}=\frac{40}{100}=40\text{\%} \\ or \\ \frac{2}{5}=0.4=40\text{\%} \end{gathered}[/tex]So the correct option is E.
shirts are 15% off. The original price of one shirt is $20. What is the total cost, in dollars, of a shirt, at the sales price, including a 10% sales tax?
The original price of the shirt is , 20 dollar.
It is given that the shirts are 15% off.
Therefore, the price of the shirt is ,
[tex]20-20\times\frac{15}{100}=17.[/tex]The price of the shirt is, 17 dollar after 15% off.
It is also given that there are 10% sales tax.
The total cost of the shirt is determined by including the sales tax in the price of the shirt after 15% off.
[tex]17+(17\times\frac{10}{100})=18.7[/tex]Thus, The total cost of shirt is calculated as, 18.7 dollar.
add or subtract : x/4 + 3/4 =
Answer:
x + 3 / 4
Explanation:
Given sin(x)=.4 and cot(x) >0 what is cos(x)?
The cotangent is given by the cosine over the sine.
If the cotangent is positive and the sine is positive, that means the cosine is also positive.
Now, in order to find the value of cos(x), we can use the following property:
[tex]\begin{gathered} \sin ^2(x)+\cos ^2(x)=1 \\ (0.4)^2+\cos ^2(x)=1 \\ 0.16+\cos ^2(x)=1 \\ \cos ^2(x)=1-0.16 \\ \cos ^2(x)=0.84 \\ \cos (x)=0.917 \end{gathered}[/tex]Find the zero for the polynomial function and give the multiplicity for each zero. State whether the graph crosses to x axis or touch the x axis and turn around, at each zero.
we have the function
f(x)=2(x-6)(x-7)^2
REmember that the zeros of the function are the values of x when the value of the function is equal to zero
In this problem
the zeros of the function are
x=6 -------> multiplicity 1 (the graph crosses to x axis)
x=7 ----- multiplicity 2 (touch the x axis and turn around)
see the attached figure to better understand the problem
cos2 0-cos 20 = sin2 0
According to Double identities
[tex]\cos (2x)=2cos^2(x)-1[/tex][tex]\begin{gathered} \cos ^2(x)-sen^2(x)=2\cos ^2(x)-1 \\ -sen^2(x)=2\cos ^2(x)-\cos ^2(x)-1 \\ -sen^2(x)=\cos ^2(x)-1 \\ 1=\cos ^2(x)+sen^2(x) \end{gathered}[/tex]What is the equation of the following graph?A. f(x) = 2(3*)OB. f(x) = (4)Oc. f(x) = 3(2)D. f(x) = 5(2") y
Given
The graph,
To find:
The equation representing the given graph.
Explanation:
It is given that,
That implies,
From the given graph,
It is clear that the curve passes through, (0,5).
Then, for x=0,
Consider the equation,
[tex]\begin{gathered} f(0)=5(2^0) \\ =5\times1 \\ =5 \end{gathered}[/tex]Which is equal to y=5.
Hence, the equation representing the above graph is,
[tex]f(x)=5(2^x)[/tex]Ethan found the spinner shownbelow and planned to use it for agame.2332453235After studying the spinner beforeusing it, Ethan correctly concludedthat the spinner was-A least likely to land on 2B least likely to land on 5C most likely to land on 3D most likely to land on 2
we have that
the number 3 appears 4 times
so
answer is
option C most likely to land on 3
11) -3(1 + 6r) = 14 - r 12) 660+6)-5=1+6v 13) -4k + 2(5k - 6) = -3k - 39 14) -16 + 5n=-71-
-3(1+6r) = 14 -r
-3-18r = 14 - r
-3-14 = -r +18r
-17 = 17r
17r/17 = -17/17
r = -1
2. Write a story that can be represented by the equation y = x + 1/4 x.Question 2 On a hot day a football team drank an entire 50-gallon cooler of water and half as much again. How much water did they drink? Create an equation to represent this situation.
y= x+ 1/4 x
Y = dependent variable
x= independent variable
Jenny has a bank account. In the first month, she deposits a certain amount of money (x), and in the month after she deposits 1/4 of that amount.
Find the total amount of money deposited (y).
Find the measurement of each side indicated and round to the nearest tenth for both triangles
a) We have a right triangle.
We have to find the value of x, which is the hypotenuse.
We can relate the angle B, the side AC and x with a trigonometric ratio as:
[tex]\begin{gathered} \sin (B)=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{AC}{AB} \\ \sin (57\degree)=\frac{10.8}{x} \\ x=\frac{10.8}{\sin (57\degree)} \\ x\approx\frac{10.8}{0.83867} \\ x\approx12.9 \end{gathered}[/tex]b) In this case, x is the adyacent side to angle A.
We can relate the sides and the angle as:
[tex]\begin{gathered} \cos (A)=\frac{\text{Adyacent}}{\text{Hypotenuse}}=\frac{AC}{AB} \\ \cos (47\degree)=\frac{x}{3} \\ x=3\cdot\cos (47\degree) \\ x\approx3\cdot0.682 \\ x\approx2.0 \end{gathered}[/tex]Answer:
a) x = 12.9
b) x = 2.0
Write the equation of the line that passes through the points (12, 4) and (22,9).
Given the following points that pass through the line:
Point A : 12,4
Point B : 22,9
Step 1: Let's determine the slope of the line (m).
[tex]\text{ m = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\text{ = }\frac{9\text{ - 4}}{22\text{ - 12}}[/tex][tex]\text{ m = }\frac{5}{10}\text{ = }\frac{1}{2}[/tex]Step 2: Let's determine the y-intercept (b). Substitute m = 1/2 and x,y = 12,4 in y = mx + b.
[tex]\text{ y = mx + b}[/tex][tex]\text{ 4 = (}\frac{1}{2})(12)\text{ + b }\rightarrow\text{ 4 = }\frac{12}{2}\text{ + b}[/tex][tex]\text{ 4 = 6 + b}[/tex][tex]4\text{ - 6 = b}[/tex][tex]\text{ -2 = b}[/tex]Step 3: Let's complete the equation. Substitute m = 1/2 and b = -2 in y = mx + b.
[tex]\text{ y = mx + b}[/tex][tex]\text{ y = (}\frac{1}{2})x\text{ + (-2)}[/tex][tex]\text{ y = }\frac{1}{2}x\text{ - 2}[/tex]Therefore, the equation of the line is y = 1/2x - 2.
two slices of dans famous pizza have 230 calories how many calories would you expect to be in 5 slices of pizza
We can answer this question, using proportions. We can see it graphically as follows:
Then, we have that 5 slices will have 575 calories.
fill in the table using the function rule y= 6x-3
Answer:
-9,-3,3,27
Step-by-step explanation:
Just multiply x by 6 and subtract 3 to that
After reading the question what would the inequality equation and the graph shade look like?
It is given that the one number is always less than the opposite of the other.
So the equation formed is y<-x
The graph is obtained as
The table shows a function. Is the function linear or nonlinear?x y0 1918 1200
By plotting the points, we get a non-linear function
The coordinates of triangle LMN are shown.YL(-1,4)XN (4, -1)M(-1, -3)What is the length of LM? Enter the answer in the box.unit(s)
Given data:
The given coordinate of L is (-1,4).
The given coordinate of M is(-1, -3).
The expression for LM length is,
[tex]\begin{gathered} LM=\sqrt[]{(-1-(-1))^2+(-3-4)^2} \\ =\sqrt[]{0+49} \\ =7 \end{gathered}[/tex]Thus, the LM length is 7 units.
3. Darren won Round 3 of the game. Sherri is wondering if she lost Round 3 by 5 points or by 25 points. Explain to Sherri how many points Darren won Round3 by and show the mathematics you used to justify your answer.4. Sherri and Darren actually played with a third player, their friend Eric. Unfortunately, Eric forgot to record the points he scored in each of the three roundsin the table.
Sherry lost the round 3 by 25 points
Explanation:Sherry's point in third round = -10
Darren's point in the third round = 15
To determine the number of points Sherry lost round 3 by, we will subtracct Sherry's point from Darren's point:
[tex]\begin{gathered} \text{Darren's point - Sherry's point} \\ =\text{ 15 - (-10)} \\ =\text{ 15 + 10} \\ =\text{ 25} \end{gathered}[/tex]Sherry lost round 3 by 25 points
A line passes through the points (7,9) and (10,1). What is its equation in point-slope form?
Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
Answer:
[tex](y - 9) = (-8/3)\, (x - 7)[/tex].
Step-by-step explanation:
If a line in a cartesian plane has slope [tex]m[/tex], and the point [tex](x_{0},\, y_{0})[/tex] is on this line, then the point-slope equation of this line will be [tex](y - y_{0}) = m\, (x - x_{0})[/tex].
The slope of a line measures the rate of change in [tex]y[/tex]-coordinates relative to the change in the [tex]x[/tex]-coordinates. If a line goes through two points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], the slope of that line will be:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}}\end{aligned}[/tex].
In this question, the two points on this line are [tex](7,\, 9)[/tex] and [tex](10,\, 1)[/tex], such that [tex]x_{0} = 7[/tex], [tex]y_{0} = 9[/tex], [tex]x_{1} = 10[/tex], and [tex]y_{1} = 1[/tex]. Substitute these values into the equation to find the slope of this line:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}} \\ &= \frac{1 - 9}{10 - 7} \\ &= \left(-\frac{8}{3}\right)\end{aligned}[/tex].
With the point [tex](7,\, 9)[/tex] as the specific point [tex](x_{0},\, y_{0})[/tex] (such that [tex]x_{0} = 7[/tex] and [tex]y_{0} = 1[/tex]) as well as a slope of [tex]m = (-8 / 3)[/tex], the point-slope equation of this line will be:
[tex]y - y_{0} = m\, (x - x_{0})[/tex].
[tex]\displaystyle y - 9 = \left(-\frac{8}{3}\right)\, (x - 7)[/tex].
Write an equation that represents a reflection in the y-axis of the graph of g(x)=|x|.
h(x)= ?
the reflection of the function g(x)=|x| in the y-axis will be h(x) = |x|
What is reflection in coordinate geometry ?
this represents the flip or mirror image of transformation about the given axis.
For every point in the plane (x, y), a 90° rotation can be described by the transformation P(x, y) → P'(-y, x). We can achieve this same transformation by performing two reflections.
Here, the given function is :
g(x)=|x|
Now, the reflection in the y-axis will be same that is :
h(x)= g(x)
h(x) = |x|
Therefore, the reflection of the function g(x)=|x| in the y-axis will be h(x) = |x|
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Use the following function rule to find (2). f(x) = (-5 - 2x)? ? f(2) = | Submit
To find f(2) you subtitute the value of x in the function by 2:
[tex]f(2)=-5-2(2)[/tex][tex]f(2)=-5-4[/tex][tex]f(2)=-9[/tex]Then, f(2) is -9For which values of A, B, and C will Ax + By = C be a horizontal line through the point (−4, 2)?
The set of values {A = 0, B = 1, and C = 2} to have a completely horizontal line.
What is a horizontal line?A horizontal line is defined as a line with slope m = 0 that is parallel to the x-axis.
A horizontal line across (-4,2) informs us of two things.
A horizontal line with slope m = 0 is parallel to the x-axis.
The line crosses the point (-4,2).
Ax + By = C has m = B/A = 0 slope and intersects point (-4,2).
Then, B = A×0 indicates that any constant A will work, and the Ax term disappears.
Ax + By = C then becomes y = C. To find C, use the point (-4,2).
⇒ C = 2
This line's equation is y = 2, and any point (x,2) matches the equation.
Therefore, the set of values {A = 0, B = 1, and C = 2} to have a completely horizontal line.
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Answer:
Step-by-step explanation:
17
TWENTY POINTS//WILL MARK BRAINLIEST
Marty graphs the hyperbola (y+2)236−(x+5)264=1 .
How does he proceed?
Drag a value, phrase, equation, or coordinates in the boxes to correctly complete the statements.
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Marty first identifies the center of the hyperbola as Response area, and that the hyperbola opens Response area. Since a = Response area, the coordinates of the vertices are Response area.
The slopes of the asymptotes of this parabola are ± Response area, and the asymptotes pass through the center of the hyperbola. The equations of the asymptotes are Response area.
Once this information is gathered, the asymptotes are graphed as dashed lines, and the hyperbola is drawn through the vertices, approaching the asymptotes.
The procedure to construct the graph of the hyperbola is described as follows:
Marty first identifies the center of the hyperbola as (-5,2), and that the hyperbola opens up and down. Since a = 6, the coordinates of the vertices are (-5, -4) and (-5, 8).The slopes of the asymptotes of this parabola are a = ± 3/4, and the asymptotes pass through the center of the hyperbola. The equations of the asymptotes are y - 2 = ± 3/4(x + 5).Hyperbola equation and graphThe equation of a vertical hyperbola with center (x*, y*) is given according to the equation presented as follows:
(y - y*)²/a² - (x - x*)²/b² = 1.
This means that the hyperbola opens up vertically, up and down.
The equation of the hyperbola in this problem is given as follows:
(y + 2)²/36 - (x + 5)²/64 = 1.
Thus the coordinates of the center are given as follows:
(-5, 2).
The numeric value of coefficient a is calculated as follows:
a² = 36 -> a = 6.
Meaning that the coordinates of the vertices of the hyperbola are given as follows:
(-5, 2 - 6) = (-5,-4).(-5, 2 + 6) = (-5,8).The slopes of the asymptotes of the parabola are given according to the rule presented as follows:
±a/b.
The coefficient b is calculated as follows:
b² = 64 -> b = 8.
Hence:
a/b = 6/8 = 3/4.-a/b = -6/8 = -3/4.Since the asymptotes pass through the center, the equation is:
y - 2 = ± 3/4(x + 5).
The graph is given by the image at the end of the answer.
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98) Cost to store: $145Markup: _?Selling price:$319.
Answer: Mark up is 220%
Cost to store : $145
selling price = $319
let x = mark up
Using the below equation
[tex]\begin{gathered} \text{Part = whole x percentage} \\ \text{part = selling price} \\ \cos t\text{ to store = whole} \\ \text{Mark up = percentage} \\ 319\text{ = 145 }\cdot\text{ x\%} \\ \text{ x\% = }\frac{319}{145}\text{ x 100\%} \\ x\text{ = 2.2 x 100\%} \\ x\text{ = 220\%} \\ \text{Therefore, the mark up is 220\%} \end{gathered}[/tex]how do i solve (4x^3 + 2x - 3) divided by (x - 3) with long division??
We want to divide 4x³ + 2x - 3 by x - 3 with the long division method
First, we rewrite the polynomial as:
4x³ + 0x² + 2x - 3
Them we divide the first term of the dividend by the highest term of the divisor:
4x²
x - 3 |4x³ + 0x² + 2x - 3
Then we multiply the divisor by this result:
| 4x²
x - 3 |4x³ + 0x² + 2x - 3
4x³ - 12x²
Now we subtract this result from the dividend:
| 4x²
x - 3 |4x³ + 0x² + 2x - 3
4x³ - 12x²
|12x² + 2x - 3
Now we can repeat all the previous steps using the last result as dividend:
| 4x² + 12x
x - 3 | 12x² + 2x - 3
12x² - 36x
|38x - 3
And we repeat these steps once more:
4x² + 12x + 38
x - 3 | 38x - 3
34x - 102
|99
The final term is the remainder
Then, we have:
4x³ + 2x - 3 = (x - 3)*(4x² + 12x + 34) + 99
Solve the Equation , I tried multiplying that 6 to the numbers in the parentheses but i couldnt get it
To start we will do what you said, multiply the 6, so:
6*(n-4)=3n
6n-6*4=3n
6n-24=3n
6n=3n+24
6n-3n=24
3n=24
n=24/3
n=8
So the answer is: 8