Kern Shipping Inc. has a requirement that all packages must be such that the combined length plus the girth (the perimeter of the cross section) cannot exceed 99 inches. Your goal is to find the package of maximum volume that can be sent by Kern Shipping. Assume that the base is a square.
a. Write the restriction and objective formulas in terms of x and y. Clearly label each.
b. Use the two formulas from part (a) to write volume as a function of x, V(x). Show all steps.
Answer:
Step-by-step explanation:
From the given information:
a)
Assuming the shape of the base is square,
suppose the base of each side = x
Then the perimeter of the base of the square = 4x
Suppose the length of the package from the base = y; &
the height is also = x
Now, the restriction formula can be computed as:
y + 4x ≤ 99
The objective function:
i.e maximize volume V = l × b × h
V = (y)*(x)*(x)
V = x²y
b) To write the volume as a function of x, V(x) by equating the derived formulas in (a):
y + 4x ≤ 99 --- (1)
V = x²y --- (2)
From equation (1),
y ≤ 99 - 4x
replace the value of y into (2)
V ≤ x² (99-4x)
V ≤ 99x² - 4x³
Maximum value V = 99x² - 4x³
At maxima or minima, the differential of [tex]\dfrac{d }{dx}(V)=0[/tex]
[tex]\dfrac{d}{dx}(99x^2-4x^3) =0[/tex]
⇒ 198x - 12x² = 0
[tex]12x \Big({\dfrac{33}{2}-x}}\Big)=0[/tex]
By solving for x:
x = 0 or x = [tex]\dfrac{33}{2}[/tex]
Again:
V = 99x² - 4x³
[tex]\dfrac{dV}{dx}= 198x -12x^2 \\ \\ \dfrac{d^2V}{dx^2}=198 -24x[/tex]
At x = [tex]\dfrac{33}{2}[/tex]
[tex]\dfrac{d^2V}{dx^2}\Big|_{x= \frac{33}{2}}=198 -24(\dfrac{33}{2})[/tex]
[tex]\implies 198 - 12 \times 33[/tex]
= -198
Thus, at maximum value;
[tex]\dfrac{d^2V}{dx^2}\le 0[/tex]
Recall y = 99 - 4x
when at maximum x = [tex]\dfrac{33}{2}[/tex]
[tex]y = 99 - 4(\dfrac{33}{2})[/tex]
y = 33
Finally; the volume V = x² y is;
[tex]V = (\dfrac{33}{2})^2 \times 33[/tex]
[tex]V =272.25 \times 33[/tex]
V = 8984.25 inches³
What sample size is needed to give a margin of error within in estimating a population mean with 95% confidence, assuming a previous sample had .
Answer:
[tex]n = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2[/tex]
The sample size needed is n(if a decimal number, round up to the next integer), considering the estimate of the proportion [tex]\pi[/tex](if no previous estimate use 0.5) and M is the desired margin of error.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is of:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
Needed sample size:
The needed sample size is n. We have that:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]M = 1.96\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]\sqrt{n}M = 1.96\sqrt{\pi(1-\pi)}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{\pi(1-\pi)}}{M}[/tex]
[tex](\sqrt{n})^2 = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2[/tex]
[tex]n = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2[/tex]
The sample size needed is n(if a decimal number, round up to the next integer), considering the estimate of the proportion [tex]\pi[/tex](if no previous estimate use 0.5) and M is the desired margin of error.
Person A leaves his home to visit his cousin, person B, who lives 243 miles away. He travels at an average rate of 46 miles per hour. One half-hour later, person B leaves her house to visit person A, traveling at an average rate of 64 miles per hour. How long after person B leaves will it be before they meet?
HELP!!!
A spherical baseball has a diameter of 5 inches and weighs 7 grams per cubic inch. What is the closest weight of the baseball rounded to the nearest gram?
Answer:
69
Step-by-step explanation:
xvusvsuvtuvqYSQY
If 3 is added to a number and the sum is multiplied by 7,gives 91 as result. Find the number.
Answer:
10
Step-by-step explanation:
10+3 = 13
13 x 7 = 91
Answer:
7(3x)=91
21x=91
x=91/21
7056 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows and the number of plants in each row
Answer:
84 rows
84 plants on each row
Step-by-step explanation:
Given
[tex]Plants = 7056[/tex]
Required
The number of rows
Since the number of rows (r) and the plants on each row (p) are the same, then the area to be planted on is:
[tex]Plants = r * p[/tex]
Where
[tex]r =p[/tex]
So, we have:
[tex]Plants = r * r[/tex]
[tex]Plants = r^2[/tex]
This gives:
[tex]7056 = r^2[/tex]
Take square roots
[tex]84 = r[/tex]
Rewrite
[tex]r = 84[/tex]
Help me pls and thanks important!!!!
Answer:
33
1/2(83-50)=25
83-x=50
x=33
I’ll give brainliest
Answer:
y = 1.19x
Step-by-step explanation:
y is the dependent variable (total cost)
x is the independent variable (number of pounds)
Consider the following confidence interval interpretation: We are 98% confident that the true mean laptop screen size is between 19.2 and 21.4 inches. Is this interpretation of a confidence interval correct or incorrect
Answer:
Incorrect
Step-by-step explanation:
This interpretation is incorrect because it states that 98% of the data is with in the confidence interval.
or 98% of the laptop have screen size between 19.2 and 21.4 inches
However, the interpretation would have been correct if it would have stated as - Value of the population mean i.e mean size of the laptop screen lies within the confidence interval.
How many 1/6 cup serving of rice and in 2/3 cup of rice
Answer:
4 serving cups
Step-by-step explanation:
Given
[tex]Serving\ cup = \frac{1}{6}[/tex]
[tex]Rice\ cup = \frac{2}{3}[/tex]
Required
The number of serving cup (n)
This is calculated by dividing the rice cup by the serving cup
[tex]n = \frac{Rice\ cup}{Serving\ cup}[/tex]
[tex]n = \frac{2/3}{1/6}[/tex]
Rewrite as:
[tex]n = \frac{2}{3} \div \frac{1}{6}[/tex]
Change to multiplication
[tex]n = \frac{2}{3} * \frac{6}{1}[/tex]
[tex]n = \frac{12}{3}[/tex]
[tex]n=4[/tex]
If you answer this you will get 100 points pls answer hurry it’s due in 20 mins
Answer:
Option B, medianStep-by-step explanation:
We see the data is not uniform. One of the prices is very high compared to the others. Since there is an outlier, the median is the best measure of central tendency.
This is the second option.
The parent function f(x) 3 to the square root of x is transformed to g(x)=2f(x-3). Which is the graph of g(x)?
Answer:
the answer is B
Step-by-step explanation:
A rectangular pyramid with a base of 9 units by 4 units and a height of 7 units.
Which is the correct calculation for the volume of the pyramid?
One-third(36)(7)= 84 units3
One-half(36)(7) = 126 units3
36(7) = 252 units3
36(7)(3) = 756 units3
The answer is A.
Hope this helps! can i have brainliest lol
Answer:
a
Step-by-step explanation:
Sue believes that the two cylinders shown in the diagram have equal volumes. Is Sue correct? Explain why or why not
Sui believes that the two cylinders have an equal volume is correct.
The volume of cylinder = Base area×Height
=(πr²)×(height)
This is also true for oblique cylinder.
Both cylinders have same height and radius also so, both cylinders have equal volume.
V=πr²h
V=π×(5)²×11.5
V=903.208m³
It can also be visualized that the oblique cylinder have circular pieces. These pieces can be solid together to form a regular cylinder with the same height (i.e. 11.4 m). So both the cylinders have equal volume.
What is a cylinder?A cylinder is a three-dimensional shape in geometry. A cylinder is round and has a top and bottom in the shape of a circle. The top and bottom are flat and always the same size.
Thus, Sue is correct, and it's true that the two cylinders shown in the diagram have equal volumes.
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Abigail is using blocks to build a tower. The blocks are 3 inches, 4 inches, and 8 inches tall. She has stack 3 blocks. How many different heights are possible for the tower?
9514 1404 393
Answer:
10
Step-by-step explanation:
Possible tower heights using 3 blocks are ...
{9, 10, 11, 12, 14, 15, 16, 19, 20, 24}
There are 10 different heights possible.
_____
Each block can be used 1, 2, or 3 times.
Using a 3 in block as the smallest, we have ...
3+3+3 = 9
3+3+4 = 10
3+3+8 = 14
3+4+4 = 11
3+4+8 = 15
3+8+8 = 19
Using a 4-in block as the smallest, we have ...
4+4+4 =12
4+4+8 = 16
4+8+8 = 20
And ...
8+8+8 = 24
A populations instantaneous growth rate is the rate at which it grows for every instant in time. Function r gives the instantaneous growth rate of a bacterial culture x hours after the start of an experiment How many hours after the experiment began was the instantaneous growth rate equal to 0? r(x)=0.01(x+2)(x^2 -9) A. 9 B. 2 C. 0 D. 3
Answer:
3
Step-by-step explanation:
r(x) = 0.01(x + 2)(x^2 - 9)
We are looking fo the value of x at which r(x) = 0.
We set the function equal to 0 and solve for x.
0.01(x + 2)(x^2 - 9) = 0
Divide both sides by 0.01. Factor x^2 - 9 as the difference of two squares.
(x + 2)(x + 3)(x - 3) = 0
x + 2 = 0 or x + 3 = 0 or x - 3 = 0
x = -2 or x = -3 or x = 3
Since we are looking for a time after the experiment started, and it started at x = 0, we discard the negative answers, and we keep x = 3.
Answer: 3
Answer:
3
Step-by-step explanation:
EDMENTUM
Round 3,872.3155 to the nearest hundred
Answer:
3,872.316
Step-by-step explanation:
Answer:
to nearest hundreds it would be: 3,900
Step-by-step explanation:
872 it closer to 900 than 800
Help?!??? What is the distance!
Trig. Ch.7.1
9514 1404 393
Answer:
100 m
Step-by-step explanation:
One side between two angles is given, so we need to find the third angle. The sum of angles in a triangle is 180°, so the third angle is ...
A = 180° -110°20' -13°20' = 56°20'
The law of sines can be used to find the length of side AB:
AB/sin(C) = BC/sin(A)
AB = BC·sin(C)/sin(A) . . . . multiply by sin(C)
AB = (360 m)sin(13°20')/sin(56°20') ≈ 99.75253 m
The distance AB across the river is about 100 m.
Solve one-sixth ÷ 6 = ___
Answer:6/6
Step-by-step explanation:
1/6 divide 6
Keep
Flip
Change
your gonna keep 1/6 as a fraction
then flip the second fraction so 6 would’ve been 6/1 but it’s gonna flip so it should look like this 1/6
after you will change the sign so the division will become multipliaction so it should look like 1/6 x 6/1 which will equal to 6/6
h(-7)=
See graph below to help solve.
two thirds of a number is negative six. find the number
Answer:
-9
Step-by-step explanation:
Two-thirds of a number is negative six. The number is -9. Let the number is x, 2/3x =-6, x=-6x3/2=-9.
In a blind ESP test, a person correctly identifies whether a tossed coin comes
up heads or tails in 63 trials out of 200. Using the normal approximation
(without the continuity correction), which of the following would you use to
calculate the probability of correctly identifying 63 or more?
Answer:
C
Step-by-step explanation:
the probability of the outcome being greater than 63 is x≥63
Names the figures in two different ways.
Answer:
hsjsjsjsjaiisisudujsjsjsjsj
If x + y =6, y +z = 7 and z + x =9, what is the average (arithmetic mean) of
x,y and z?
Answer:
[tex]\frac{22}{3}[/tex]
Step-by-step explanation:
x + y + y + z + z + x = 2x + 2y + 2z = 22
x + y + z = 22
22 ÷ 3 = [tex]\frac{22}{3}[/tex]
PLEASE HELP !!! WILL MARK BRAINLIEST TO WHOEVER GETS IT RIGHT !!
Answer:
-2 and 0
Step-by-step explanation:
EZ
What's the Error? Explain the error. Find the correct solution.
12-(-8)= 4
Step-by-step explanation:
12-(-8)=4
12+8=20
because - - sign +
plzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz help me i really do need the help
Each of the problems below was solved incorrectly, for each problem, find the mistake in the work/ answer. Explain what the mistake is, and find the correct answer.
Explain the mistake:
Find the correct answer(equation):
2. Find the value of x
Explain the mistake:
Find the correct answer(equation):
3. Find the value of x
Explain the mistake:
Find the correct answer(equation):
Question 1
The mistake is that vertical angles are congruent, and don't always add up to 180 degrees.[tex]5x=100 \longrightarrow x=20[/tex]Question 2
Angles that add to form a right angle add to 90 degrees, not 180 degrees.[tex]3x+39=90 \longrightarrow 3x=51 \longrightarrow x=17[/tex]Question 3
Angles that add to form a right angle add to 90 degrees, not 180 degrees.[tex]3x+39=90 \longrightarrow 3x=51 \longrightarrow x=17[/tex]A science teacher wrote the table of values below.
Amount of Hydrogen vs. pH
pH, f(x)
Amount of Hydrogen, X
(in moles per liter)
1
10
1
2
1
100
3
3
1
1,000
4
1
10.000
-
5
1
100,000
Which function models the data in the table?
Answer:
B
Step-by-step explanation:
Took quiz
The logarithmic function for the teacher who wrote the table of values with amount of hydrogen and pH level is given by f ( x ) = log ( 1/x )
What is Logarithm?The power to which a number must be increased in order to obtain another number is known as the logarithm. A power is the opposite of a logarithm. In other words, if we subtract an exponentiation from a number by taking its logarithm
The properties of Logarithm are :
log A + log B = log AB
log A − log B = log A/B
log Aⁿ = n log A
Given data ,
Let the function be represented as f ( x )
Now , the values of x are
x = { 1/10 , 1/100 , 1/1000 , 1/10000 , 1/100000 }
where x represents the amount of Hydrogen in moles per liter
And the values of f ( x ) = { 1 , 2 , 3 , 4 , 5 }
where y represents the pH level
So , when x = 1/10 , f ( x ) = 1
And , the logarithmic equation is given as
f ( x ) = log ( 1/x )
when x = 1/1000
f ( 1/1000 ) = log ( 1/1/1000 )
f ( 1/1000 ) = log ( 1000 )
f ( 1/1000 ) = 3
Therefore , the value of f ( x ) is log ( 1/x )
Hence , the logarithmic equation is f ( x ) = log ( 1/x )
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Identify the relationship between sampling error and sample size.
Answer:
as the sample size increases, the margin of error decreases
The hypotenuse of a right triangle is 13 ft long. The longer leg is 7 ft longer than the shorter leg. Find the side lengths of the triangle.
Answer:
The shorter leg is five feet, the longer leg is 12 feet, and the hypotenuse is 13 feet.
Step-by-step explanation:
Let the shorter leg be x.
Since the longer leg is seven feet longer than the shorter leg, the length of the longer leg can be modeled by (x + 7).
Since the triangle is a right triangle, we can use the Pythagorean Theorem, given by:
[tex]a^2+b^2=c^2[/tex]
Where a and b are the side lengths and c is the hypotenuse.
The hypotenuse is 13 and the legs are x and (x + 7). Substitute:
[tex](x)^2+(x+7)^2=(13)^2[/tex]
Square:
[tex]x^2+x^2+14x+49=169[/tex]
Simplify:
[tex]2x^2+14x-120=0[/tex]
We can divide both sides by two:
[tex]x^2+7x-60=0[/tex]
Factor:
[tex](x-5)(x+12)=0[/tex]
Zero Product Property:
[tex]x-5=0\text{ or }x+12=0[/tex]
Solve for each case:
[tex]x=5\text{ or } x=-12[/tex]
Since lengths cannot be negative, we can ignore the negative answer. So, our only solution is:
[tex]x=5[/tex]
The shorter leg is five feet, the longer leg will be (5 + 7) or 12 feet. And the hypotenuse is 13 feet as given.