There is no Graph to go in response to this question
A particle moves along the curve y=7 x 2+4y=7 x 2+4 in such a way that its xx-coordinate is changing at a rate of −5−5 centimeters per second. At what rate is the particle's yy-coordinate changing when the particle is at the point where x=1x=1?
Answer:
The y-coordinate is changing by the rate of -70 cm per sec.
Step-by-step explanation:
Given equation,
[tex]y = 7x^2 + 4[/tex]
Differentiating with respect to time (t),
[tex]\frac{dy}{dt}=14x \frac{dx}{dt}[/tex]
We have,
[tex]\frac{dx}{dt}=-5\text{ cm per sec}, x = 1[/tex]
[tex]\frac{dy}{dt} = 14(1)(-5)=-70\text{ cm per sec}[/tex]
Ben paid 45 dollers for an old giutar than resold it after marking up the price 15 persent he used 36 persent of the money he got fornthe giutar tpo buy books how much money did ben spend on books
Answer:
18.63
Step-by-step explanation:
45*115%= 45*1.15 = 51.75
51.75*0.36 = 18.63
Answer:
18.63 dollars.
Step-by-step explanation:
The selling price was 45 + 15% or 45
= 45 + 0.15*45
= 51.75 dollars.
36% of 51.75
= 0.36 * 51.75
= 18.63 dollars which he had to spend on books.
In Andrew’s Furniture Shop, he assembles both bookcases and TV stands. Each type of furniture takes him about the same time to assemble. He figures he has time to make at most 18 pieces of furniture by this Saturday. The materials for each bookcase cost him $20.00 and the materials for each TV stand cost him $40.00. He has $600.00 to spend on materials. Andrew makes a profit of $60.00 on each bookcase and a profit of $100.00 for each TV stand. Find how many of each piece of furniture Andrew should make so that he maximizes his profit.
Answer:
6 bookcases12 TV standsStep-by-step explanation:
Given Andrew has $600 for materials and can make 18 pieces of furniture, you want to know the number of each kind that maximizes profit if each bookcase costs $20 and gives $60 profit, while each TV stand costs $40 and gives $100 profit.
SetupIf x and y represent the numbers of bookcases and TV stands Andrew builds, respectively, then he wants to ...
maximize 60x +100y
subject to ...
x + y ≤ 1820x +40y ≤ 600SolutionThe attached graph shows the solution space for these constraints. The profit is maximized at the vertex of the space where the profit function line is farthest from the origin. Andrew maximizes his profit by building ...
6 bookcases12 TV standsAndrew needs to solve a linear programming problem to find how many bookcases and TV stands he should manufacture for optimal profit. This is done by setting up and solving inequalities representing Andrew's time and material cost constraints, graphing the feasible region, and finding the point(s) in this region that yield the highest profit.
Explanation:This question deals with the topics of linear programming and profit maximisation. Here, Andrew has to decide how much of each type of furniture, bookcases or TV stands, he should produce to maximise profit while considering time and material cost constraints.
From the given conditions, we get two inequalities. The first related to time says that the total number of bookcases and TV stands is less than or equal to 18: let bookcases be x, TV stands be y, thus we have x + y <= 18. The second involving the cost of material says that the total cost spent on materials for both products does not exceed $600: thus, we also have 20x + 40y <= 600.
You can graph these inequalities on the x-y plane to get a visual representation of the possibilities.
Finally, to find the optimal solution (i.e., the highest profit), you calculate the profit function P = 60x + 100y for each point in the feasible region and select the point that provides the highest profit.
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Omar picks 7 pounds of cherries . He divides them into containers that holds one whole 1/4 pounds of blue berries each. How many containers can he fill?
Answer:
Omar can fill 28 containers
Step-by-step explanation:
Omar have 7 pounds of cherries, each pound need 4 containers because each one only hold of [tex]\frac{1}{4}\\[/tex] pound, now we multiplicate 7 pounds with 4 container for each one and we get 28 containers.
[tex]Number containers = \frac{7 pounds}{\frac{1}{4}pounds } = 28[/tex]
A deck is shuffled and two cards are dealt. (a) Find the chance that the second card is a heart given the first card is a heart. (b) Find the chance that the first card is a heart and the second card is a heart.
Answer: a) 12/51
b) 3/51
Step-by-step explanation:
we are assuming the cards are dealt without replacement
a) given that the first card is heart, we are left with 12 hearts and 51 cards in total
Therefore, the probability that the second card is heart is:
P2 = 12/51
b) probability that the two cards are hearts is given by:
P = 13/52 * 12/51
P = 3/51
(a) The chance that the second card is a heart given the first card is a heart is [tex]\(\frac{12}{51}\)[/tex].
(b) The chance that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].
(a) To find the probability that the second card is a heart given that the first card is a heart, we use conditional probability. There are 13 hearts in a standard deck of 52 cards. Once the first heart is drawn, there are 12 hearts left and the total number of cards left is 51. The probability of drawing a heart as the second card, given that the first card is a heart, is the number of remaining hearts divided by the total number of remaining cards.
[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{12}{51} \][/tex]
This simplifies to:
[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{4}{17} \][/tex]
(b) To find the probability that both the first and second cards are hearts, we multiply the probability of drawing a heart first by the probability of drawing a heart second given that the first card is a heart. The probability of drawing a heart first is [tex]\(\frac{13}{52}\)[/tex], which simplifies to [tex]\(\frac{1}{4}\)[/tex]. We already calculated the probability of drawing a heart second given a heart first as [tex]\(\frac{12}{51}\) or \(\frac{4}{17}\)[/tex].
[tex]\[ P(\text{first card is a heart and second card is a heart}) = P(\text{first card is a heart}) \times P(\text{second card is a heart | first card is a heart}) \][/tex]
[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{4} \times \frac{4}{17} \][/tex]
[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{17} \][/tex]
Thus, the probability that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].
A piece of fabric is 7/9 yard long.A piece of ribbon is 2/9 yard long.How many more yards of ribbon do you need to have equal lengths of fabric and ribbon
Answer:
5/9 yards
Step-by-step explanation:
Just subtract 2/9 from 5/9 to find the difference, which is the answer.
Debby is making pizzas. She needs to choose among three bags of shredded mozzarella. One contains 8 ounces and costs $1.59. One contains 12 ounces and costs $2.49. One contains 16 ounces and costs $3.29. If Debby needs 48 ounces of cheese for her pizzas, how many of which type of bag should she buy and what will be the cost?
Answer:
6 of the 8 oz one
Step-by-step explanation:
you need to buy 6 of the first to get 48 oz, and the price would be 9.54. you need 4 for the next with the price at 9.96. you need 3 for 16 oz and the price is 9.89. compare all of them and you get 9.54 as the lowest, which was the 8 oz one
On a field trip, students ate 3/10 of a box of oranges.Altogether they ate 6 pounds of oranges. How many pounds of oranges were in a full box?Why each tenth of the model is 2 pounds?
Answer:
Step-by-step explanation:
Let x = the number of pounds of oranges in the full box.
On a field trip, students ate 3/10 of a box of oranges. This means that the students ate 3/10 × x = 3x/10 pounds of oranges.
Altogether they ate 6 pounds of oranges. This means that
3x/10 = 6
3x = 6×10
3x = 60
x = 60/3 = 20
The full box contained 20 pounds of oranges
Each tenth of the model is 2 pounds because a tenth of 20 pounds is 20/10 = 2 pounds
What is the value of sin C ?
A. 8/17
B. 1/58
C. 15/17
D. 8/15
Answer: option A is the correct answer
Step-by-step explanation:
The given triangle is a right angle triangle. This is because one of the angles is 90 degrees. The sum of the other two angles is 90 degrees. To determine sin C, we will apply trigonometric ratio
From the dimensions given and taking C as the reference angle,
Hypotenuse = AC = 17
Adjacent side = BC = 15
Opposite side = AB = 8
Sin# = opposite side / hypotenuse
Since # = C
SinC = 8/17
The value of angle C can be derived by finding Sin^-1(8/17)
A company makes auto batteries. They claim that 86% of their LL70 batteries are good for 70 months or longer. Assume that this claim is true. Let p be the proportion in a random sample of 80 such batteries For a populations that are good for 70 months or more.
What is the probability that this sample proportion is within 0.03 of the population proportion?
Answer:
The probability that the sample proportion of 80 LL70 batteries is within 0.03 of the population proportion is 0.44
Step-by-step explanation:
Sample proportion being within margin, or margin of error (ME) around the mean can be found using the formula
ME=[tex]\frac{z*\sqrt{p*(1-p)}}{\sqrt{N} }[/tex] where
z is the corresponding statistic of the probability that the sample proportion is within the 0.03 of the population proportionp is the claimed proportion (86% or 0.86) N is the sample size (80)Then 0.03=[tex]\frac{z*\sqrt{0.86*0.14}}{\sqrt{80} }[/tex] from this we get:
z≈0.773 and the p(z)≈0.439
Therefore, the probability that the sample proportion is within 0.03 of the population proportion is 0.44
Please help!
Options for ♣:
definition of adjacent angles
definition of angle bisector
definition of congruence
vertical angles are congruent
Options for ♦:
AAS
ASA
SAS
SSS
Answer:
The proof with the statement is given below.
Step-by-step explanation:
Given:
Construction of angle bisector i.e SP is the bisector of angle RPQ.
To prove:
Δ PWS ≅ Δ PXS
Proof:
In Δ PWS and Δ PXS
∠ WPS ≅ ∠ XPS …………..{definition of angle bisector}
SP ≅ SP .......…….{Reflexive property}
PW ≅ XP ……....….{definition of ≅}
Δ PWS ≅ Δ PXS …...........{Side-Angle-Side test i .e SAS}
Angle bisector: A ray divides angle into two equal measures then the ray is called as angle bisector of the bisected angle.
In the construction SP is the bisector of the angle ∠ RPQ
SAS: This test is to prove the triangle congruent when two sides are congruent and angle between that sides should be congruent. then we can say the triangle is congruent by side angle side test.
What is the slope of the line?
5
-5
1/5
-1/5
Answer: Slope is 1/5
Step-by-step explanation:
Slope, m is expressed as
Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis
change in the value of y = y2 - y1
Change in value of x = x2 -x1
y2 = final value of y
y 1 = initial value of y
x2 = final value of x
x1 = initial value of x
From the graph given, we would pick points for y2 and a corresponding x1 and also pick y1 and a corresponding x1
y2 = 4
y 1 = 3
x2 = 5
x1 = 0
Slope = (4-3)/(5-0) = 1/5
Answer:
[tex]\frac{1}{5}[/tex]
Step-by-step explanation:
→The slope of the line given on the graph is [tex]\frac{1}{5}[/tex].
→You can tell because following the rise over run, the line goes up 1 unit, then to the right, 5 units.
→Since it is going to the right, it stays positive. However, if it were to go left, the 5 would be negative.
A __________ describes the range and relative likelihood of all possible values for a random variable.a. probability mass function of an event.b. density function.c. probability distribution for a random variable.d. probability
Answer:
Option C: Probability distribution for a random variable
Step-by-step explanation:
A probability distribution for a random variable describes the range and relative likelihood of all possible values for a random variable.
The probability distribution of a random variable is explained as a list of all the possible values of the variable and their probabilities. These probabilities sum up to 1.
Final answer:
The correct term is ' (c) probability distribution for a random variable,' represented by a probability density function for continuous random variables or a probability distribution function for discrete random variables.
Explanation:
The term that describes the range and relative likelihood of all possible values for a random variable is a (c) probability distribution for a random variable. For a continuous random variable, this is represented by a probability density function (pdf), which shows the likelihood of any given value or range of values. The pdf is depicted graphically, and the area under the curve represents the probability for a given range of values. The total area under the pdf curve is always one, signifying that the sum of all probabilities is one. Also, for a discrete random variable, the probability distribution function (PDF) lists all possible values and their associated probabilities, following the rules that each probability is between zero and one inclusive, and the sum of all probabilities equals one.
You have a $30 gift card from Amazon that you use to purchase two T-shirts for a total of $25. The sales tax is 8%. Will your gift card cover the cost? Write an expressions and solve. Explain your answer.
Answer:
Step-by-step explanation:
Total cost of the two T-shirts that you purchased is $25. The sales tax on the T-shirts is 8%. This means that the additional amount that you will pay on the T shirt is
25/100 × 25 = 0.25 × 25 = $6.25
The new cost of the T shirt would be the sum of the original cost and the value of the sales tax. Therefore,
New cost = 25 + 6.25 = $31.25
You have a $30 gift card from Amazon. The new cost of the shirts is higher than that value of the gift card. So the gift card will not cover the cost of the T shirts
To determine if the gift card will cover the cost of the two T-shirts, calculate the sales tax and subtract it from the total cost. The gift card will cover the cost if the remaining balance is positive.
Explanation:To determine if your gift card will cover the cost of the two T-shirts, you need to calculate the cost of the sales tax and subtract it from the total cost of the T-shirts. First, convert the sales tax percentage to decimal form by dividing it by 100 (8% becomes 0.08). Then, multiply the total cost of the T-shirts (which is $25) by the sales tax rate to find the amount of sales tax. Multiply $25 by 0.08 to get $2. Next, subtract the sales tax amount from the total cost of the T-shirts: $25 - $2 = $23. Finally, determine if the gift card covers the cost by comparing it to the final cost: $30 - $23 = $7.
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A pilot has just started on the glide path for landing at an airport with a runway of length 9000 feet. The angles of depression from the plane to the ends of the runway are 17.5 degrees and 18.5 degrees.
(a) Draw a diagram that visually represents the problem
(b) Find the air distance the plane must travel until touching down on the near end of the runway.
(c) Find the ground distance the plane must travel until touching down
(d) Find the altitude of the plane when the pilot begins the descent
Answer:
a. See attachments
B. 163630.2ft
C. 147056.70ft
D. 49205.4ft
Step-by-step explanation:
Check attachments for details
Angle a.. Angle of dep to beginning of runway
Angle b... Angle of del to end of runway
AD... Altitude of plane
DB..... Ground distance before touch down
AC... Air distance travelled until touch down on the near end of runway
The sum of the first ten terms of a particular arithmetic sequence is four times the sum of the first five terms of the sequence. What is the ratio of the first term to the second term? Express your answer as a common fraction.
In the given arithmetic sequence, the ratio of the first term to the second term is 1:2.
Explanation:The problem is based on the properties of the arithmetic sequence. To solve it, we use the formula for the sum of an arithmetic sequence: S = n/2*(a + l), where n is the number of terms, a is the first term, and l is the last term.
From the question, we know that 4 times the sum of the first five terms equals the sum of first ten terms. Therefore, 4 * (5/2 * (a + a + 4d)) = 10/2 * (a + a + 9d), where d is the common difference. Simplifying, we find that the ratio of the first term a to the second term (a + d) is 1:2.
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A monopolist faces the following demand curve, marginal revenue curve, total cost curve and marginal cost curve for its product: Q = 200 - 2P MR = 100 - Q TC = 5Q MC = 5 Refer to Scenario 10.2. What is the profit maximizing level of output?
Answer:
Q=95, P= 52.5
Step-by-step explanation:
The profit maximizing level of output in monopolies is reached when the marginal cost is equal to the marginal revenue. This is also the profit maximizing rule in perfect competition, the difference between both is that is perfect competition the marginal revenue is equal to the price while in monopolies, the demand curve is often above the marginal revenue curve, then the actual price (defined by the demand curve) is often higher than the marginal revenue price.
For this problem the profit maximizing level of output is:
MC=MR
5=100-Q
Q=95
Because monopolies decide the selling price based on the demand curve, you should replace this quantity in the demand curve equation:
95=200-2P
95-200/2=-P
P= 52.5
When output is set so that marginal revenue and marginal cost are equal in this economics dilemma, the monopolist will make the most money. That would be at a manufacturing level of 95 units in this instance.
Explanation:When establishing output, a monopolist in economics attempts to maximize profit by ensuring that Marginal Cost (MC) and Marginal Revenue (MR) are equal. We must assign MR to MC in this case given that MR = 100 - Q and MC = 5. Therefore, 100 - Q = 5, giving Q=95. In order to maximize its profit, the monopolist should create 95 units of the good.
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A box is packed with 18 cans of cola. The radius of the base of one can of cola is 1 inch, and the height is 5 inches. The length of the box is 12 inches, the width is 6 inches, and the height is 5 inches. In cubic inches, how much empty space is left inside the box?
Answer:
[tex]77.4\ in^3[/tex]
Step-by-step explanation:
we know that
To find out how much empty space is left inside the box, subtract the volume of 18 cans of cola from the volume of the box
step 1
Find the volume of the box
The volume of the box is equal to
[tex]V=LWH[/tex]
substitute the given values
[tex]V=(12)(6)(5)[/tex]
[tex]V=360\ in^3[/tex]
step 2
Find the volume of the can of cola
The volume of a cylinder is equal to
[tex]V=\pi r^{2} h[/tex]
we have
[tex]r=1\ in[/tex]
[tex]h=5\ in[/tex]
substitute
[tex]V=\pi (1)^{2} (5)[/tex]
[tex]V=5\pi\ in^3[/tex]
Multiply by 18 (18 cans of cola)
[tex]V=(18)5\pi=90\pi\ in^3[/tex]
step 3
Find how much empty space is left inside the box
[tex]V=(360-90\pi)\ in^3[/tex] ---> exact value
assume
[tex]\pi =3.14[/tex]
[tex]V=360-90(3.14)=77.4\ in^3[/tex]
Answer:
77.4
Step-by-step explanation:
The numerator of a fraction is 15 less than twice its denominator. If the numerator is increased by 5 and the denominator is increased by 7, the new fraction will be equal to 2/3. What is the original fraction?
Answer:
7/11
Step-by-step explanation:
Let d represent the original denominator. Then the original numerator is ...
2d-15
The new numerator is ...
(2d-15) +5
and the new denominator is ...
d+7
The ratio of these is 2/3, so we have ...
[tex]\dfrac{2d-15+5}{d+7}=\dfrac{2}{3}\\\\3(2d-10)=2(d+7) \quad\text{cross multiply}\\\\4d=44 \quad\text{add 30-2d}\\\\d=11\\\\2d-15=2(11)-15=7[/tex]
The original fraction is 7/11.
Answer:.
Step-by-step explanation:
.
The width of a small playground 3 meters less than the length of the playground. If the area of the playground 54 square meters. What are the dimensions of the playground.
Answer: Length = 9 meters
Width = 6 meters
Step-by-step explanation:
The diagram of the playground is shown in the attached photo
The width of a small playground 3 meters less than the length of the playground.
Length of playground = L meters
Width of playground = (L-3) meters
The area of the playground is 54 square meters.
Area = L × W
54 = L(L - 3)
54 = L^2 - 3L
L^2 - 3L - 54 = 0
L^2 + 6L - 9L- 54 = 0
L(L+6) - 9(L+6) = 0
(L-9)(L+6) = 0
L -9 = 0. or L+6 = 0
L = 9 or L = -6
L cannot be negative so, the length of the playground is 9 meters
The width of the playground is
L-3 = 9-3 = 6 meters
A satellite in a circular orbit 1250 kilometers above the Earth makes one complete revolution every 110 minutes. Assuming that Earth is a sphere of radius 6378 kilometers,
what is the linear speed (in kilometers per minute) of the satellite?
What is the linear speed in kilometers per hour, in miles per hour?
Final answer:
The linear speed of the satellite is approximately 434.71 km/min. In kilometers per hour, it is approximately 26082.6 km/hr. In miles per hour, it is approximately 16206.26 miles/hr.
Explanation:
To find the linear speed of the satellite, we need to calculate the circumference of the circular orbit.
The radius of the orbit is the sum of the radius of the Earth and the altitude of the satellite:
Radius of orbit = Radius of Earth + Altitude of satellite = 6378 km + 1250 km = 7628 km
The circumference of a circle is given by the formula:
Circumference = 2π * Radius
Substituting the radius of the orbit into the formula:
Circumference = 2π * 7628 km ≈ 47818.16 km
In 110 minutes, the satellite completes one revolution around the Earth. Therefore, its linear speed is:
Linear speed = Circumference / Time taken = 47818.16 km / 110 minutes ≈ 434.71 km/min
To convert the linear speed from kilometers per minute to kilometers per hour, multiply by 60:
Linear speed = 434.71 km/min * 60 min/hr ≈ 26082.6 km/hr
To convert the linear speed from kilometers per hour to miles per hour, divide by the conversion factor of 1.60934:
Linear speed = 26082.6 km/hr / 1.60934 ≈ 16206.26 miles/hr
answer correcly / explain a lil.
match them....
Answer:
[tex]\displaystyle \boxed{66 \times 7}\:7(60 + 6)[/tex]
[tex]\displaystyle \boxed{97 \times 4}\:4(100 - 3)[/tex]
[tex]\displaystyle \boxed{8(4 + 2)}\:32 + 16 = 48[/tex]
[tex]\displaystyle \boxed{5(9 - 6)}\:(5 \times 9) - (5 \times 6)[/tex]
[tex]\displaystyle \boxed{3(4 + 7)}\:(3 \times 4) + (3 \times 7)[/tex]
Step-by-step explanation:
According to the Order of Operations [GEMS\BOMDAS\PEMDAS etc.], you evaluate everything in parentheses first before preceding with your Division & Multiplication and Subtraction & Addition. When you do this, you will know exactly which expression corresponds with its Distributive Property expression.
I am joyous to assist you anytime.
A company makes wax candles in the shape of a cylinder. Each candle has a radius of 2 inches and a height of 7 inches. How much wax will the company need to make 210 candles?
Answer:
18,471.6 cubic inches of wax will be needed
Step-by-step explanation:
We want the volume that will be required for 210 candles. We first find the volume of 1 candle by using volume of cylinder formula. Then multiply that answer by 210 to find volume of wax needed to make 210 such candles.
Volume of Cylinder is given by the formula:
[tex]V=\pi r^2 h[/tex]
Where
r is the radius
h is the height
Given,
r = 2 in
h = 7 in
We substitute and find 1 candle volume:
Volume of 1 candle = [tex]\pi r^2 h = \pi (2)^2 (7) = 87.96[/tex]
Hence,
Volume of 210 candles = 87.96 * 210 = 18,471.6 cubic inches
Answer:
18,471.6
Step-by-step explanation:
What value of x is in the solution set of 3(x-4) > 5x + 2?
Answer:
x < -7
Step-by-step explanation:
Eliminate parentheses:
3x -12 > 5x +2
-14 > 2x . . . . . . . . . add -2-3x
-7 > x . . . . . . . . . . . divide by 2
Any value of x less than -7 is in the solution set.
Jacob went on a bike ride. After 10 miles he got a flat tire and had to jog back home. He jogs 5 mph slower than he bikes, so the jog took 1 hour longer than the bike ride. At what rate did he travel each way?
Answer: He traveled 10 km/hr through bike and 5km/hr by jogging.
Step-by-step explanation:
Let the speed of bike be 'x'.
Let the speed of his jogging be 'x-5'.
Distance covered = 10 miles
So the jog took 1 hour longer than the bike ride.
According to question, we get that
[tex]\dfrac{10}{x-5}-\dfrac{10}{x}=1\\\\10\dfrac{x-x+5}{x(x-5)}=1\\\\\dfrac{50}{x^2-5x}=1\\\\50=x^2-5x\\\\x^2-5x-50=0\\\\x^2-10x+5x-50=0\\\\x(x-10)+5(x-10)=0\\\\(x+5)(x-10)=0\\\\x=10\ km/hr[/tex]
Hence, he traveled 10 km/hr through bike and 5km/hr by jogging.
Jay is letting her bread dough rise. After three hours, her bread dough is \dfrac{11}{5} 5 11 start fraction, 11, divided by, 5, end fraction of its original size.
Answer:
Jay's bread size is 220% of the original size.
Step-by-step explanation:
The question is incomplete.
The complete question is as follows.
Jay is letting her bread rise. After 3 hours,her bread is at 11/5 of its original size. What percent of its original size is jays bread dough?
Solution:
Let the original bread size be = [tex]100[/tex] units
After 3 hours the bread rises = [tex]\frac{11}{5}[/tex] of the original size
New size of bread = [tex]\frac{11}{5}\times 100=220[/tex] units
Percent of original size the new bread is
⇒ [tex]\frac{New\ bread\ size}{Original\ bread\ size}\times 100[/tex]
⇒ [tex]\frac{220}{100}\times100[/tex]
⇒ [tex]220\%[/tex]
Please help !!! I’ll mark brainliest
A) State the chain rule for integration
Ans. The chain rule for integration is also known as " Integration by substitution "
Integration by substitution is taken in order to make integration solve easily in few steps.
For, [tex]I = \int\limits (x+2)^{2} \,dx[/tex]
Instead of expanding term [tex](x+2)^{2}[/tex]
With substitution of [tex]u = (x+2) [/tex] and [tex]du= 1 dx [/tex]
We simplified the integration as
[tex]I = \int\limits (u)^{2} \, du[/tex]
[tex]I = \frac{(u)^{3}}{3}+C[/tex]
By replacaing value of u=x+2
[tex]I = \frac{(x+2)^{3}}{3}+C[/tex]
B) State the rule of differentiation for the sine function.
Ans. We know that [tex]\frac{d}{dx}Sinx dx = Cosx [/tex]
C) Find the indefinite integral using substitution.
Ans.
Given, [tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]
Take y = Sin14x
Differentiating both side
[tex]dy=14Cos14x dx [/tex]
[tex]\frac{dy}{14} = Cos14x\, dx[/tex]
Substituting values in integration,
[tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]
[tex]I = \int\limits {\frac{1}{y} } \,\frac{dy}{14} [/tex]
[tex]I = \frac{1}{14}\int\limits {\frac{1}{y} } \,dy [/tex]
[tex]I = \frac{1}{14} lny + C [/tex]
Replacing values in the integration
[tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]
D)Check your work by taking a derivative of your answer from part C.
Ans.
Answer for Part C is [tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]
Differentiating the answer
we get,
[tex]=\frac{1}{14}\frac{d}{dx}[ ln(Sin14x) + C]\\=\frac{1}{14}\frac{1}{Sin14x} \frac{d}{dx}(Sin14x)+ \frac{d}{dx}C\\=\frac{1}{14}\frac{1}{Cos14x}(14Cos14x)\\=\frac{Cos14x}{Sin14x} \\ =I[/tex]
Help asap please!!!!
A prism is made of cubes which measures 1/4 of an inch on one side. What is the volume?
A. 40 cubic in
B. 11/4 cubic in
C. 5/8 cubic in
D. 10 cubic in
Answer:
b
Step-by-step explanation:
A 27-inch by 72-inch piece of cardboard is used to make an open-top box by removing a square from each corner of the cardboard and folding up the flaps on each side. What size square should be cut from each corner to get a box with the maximum volume? Enter the area of the square and do not include any units in your answer.
Answer:
36
Step-by-step explanation:
Given:
Length of the cardboard = 27 inches
Width of the cardboard = 72 inches.
Let "x" be side of the square which is cut in each corner.
Now the height of box = "x" inches.
Now the length of the box = 27 - 2x and width = 72 - 2x
Volume (V) = length × width × height
V = (27 - 2x)(72 - 2x)(x)
[tex]V= (1944 -144x -54x + 4x^2)x\\V = (4x^2 - 198x +1944)x\\V = 4x^3 -198x^2 +1944x[/tex]
Now let's find the derivative
V' = [tex]12x^2 - 396x + 1944[/tex]
Now set the derivative equal to zero and find the critical points.
[tex]12x^2 - 396x + 1944[/tex] = 0
12 ([tex]x^2 - 33x + 162[/tex]) = 0
Solving this equation, we get
x = 6 and x = 27
Here we take x = 6, we ignore x = 27 because we cannot cut 27 inches since the entire length is 27 inches.
So, the area of the square = side × side
= 6 inches × 6 inches
The area of the square = 36 square inches.
Solve the following exponential equation by taking the natural logarithm on both sides. Express the solution in terms of natural logarithms Then. use a calculate obtain a decimal approximation for the solution. e^2 - 4x = 662
What is the solution in terms of natural logarithms?
The solution set is { }.
(Use a comma to separate answers as needed. Simplify your answer Use integers or fractions for any numbers in expression).
What is the decimal approximation for the solution?
The solution set is { }.
(Use a comma to separate answers as needed. Round to two decimal places as needed.)
Answer:
[tex]-\frac{ln(662)-2}{4}[/tex]
{-1.12}
Step-by-step explanation:
[tex]e^{2 - 4x} = 662[/tex]
Solve this exponential equation using natural log
Take natural log ln on both sides
[tex]ln(e^{2 - 4x}) = ln(662)[/tex]
As per the property of natural log , move the exponent before log
[tex]2-4x(ln e) = ln(662)[/tex]
we know that ln e = 1
[tex]2-4x= ln(662)[/tex]
Now subtract 2 from both sides
[tex]-4x= ln(662)-2[/tex]
Divide both sides by -4
[tex]x=-\frac{ln(662)-2}{4}[/tex]
Solution set is {[tex]x=-\frac{ln(662)-2}{4}[/tex]}
USe calculator to find decimal approximation
x=-1.12381x=-1.12
- In terms of natural logarithms: [tex]\( \{ \ln(2) \} \)[/tex]
- In decimal approximation: [tex]\( \{ 0.69 \} \)[/tex] (rounded to two decimal places)
To solve the exponential equation [tex]\( e^2 - 4x = 662 \),[/tex] we can follow these steps:
Step 1: Isolate the exponential term by subtracting 2 from both sides:
[tex]\[ e^2 - 2 = 662 \][/tex]
Step 2: Divide both sides by -4 to isolate ( x ):
[tex]\[ -4x = 660 \][/tex]
Step 3: Divide both sides by -4 to solve for ( x ):
[tex]\[ x = -\frac{660}{4} \][/tex]
[tex]\[ x = -165 \][/tex]
Now, let's express the solution in terms of natural logarithms:
[tex]\[ x = -165 \][/tex]
To obtain a decimal approximation for the solution, we can use a calculator. Substituting [tex]\( x = -165 \)[/tex] back into the original equation:
[tex]\[ e^2 - 4(-165) = 662 \][/tex]
[tex]\[ e^2 + 660 = 662 \][/tex]
[tex]\[ e^2 = 2 \][/tex]
Now, take the natural logarithm of both sides:
[tex]\[ \ln(e^2) = \ln(2) \][/tex]
[tex]\[ 2\ln(e) = \ln(2) \][/tex]
[tex]\[ 2 = \ln(2) \][/tex]
So, the solution in terms of natural logarithms is [tex]\( x = \ln(2) \).[/tex]
The decimal approximation for [tex]\( x = \ln(2) \)[/tex] is approximately [tex]( x \approx 0.69315 \).[/tex]
Therefore, the solution set is:
- In terms of natural logarithms: [tex]\( \{ \ln(2) \} \)[/tex]
- In decimal approximation: [tex]\( \{ 0.69 \} \)[/tex] (rounded to two decimal places)