If 300 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters
Answers
[tex]\bf V(x)=x^2\left( \cfrac{300-x^2}{4x} \right)\implies V(x)=x^2\left( \cfrac{75}{x}-\cfrac{x}{4}\right) \\\\\\ V(x)=75x-\cfrac{x^3}{4}\implies \cfrac{dV}{dx}=75-\cfrac{1}{4}\cdot 3x^2\implies \cfrac{dV}{dx}=75-\cfrac{3x^2}{4} \\\\\\ \cfrac{dV}{dx}=\cfrac{300-3x^2}{4}\impliedby \textit{now, let's set the derivative to 0} \\\\\\ 0=\cfrac{300-3x^2}{4}\implies 0=300-3x^2\implies 3x^2=300\implies x^2=100 \\\\\\ x=\pm\sqrt{100}\implies x=\pm 10[/tex]
now, if you do a first-derivative test on +10, say check 9.99 and 10.01 for example, you'll notice you'd get + and - value respectively, meaning is a maximum.
Related Questions
The Mississippi River is about 4,000 kilometers long. An M&M is about 1 centimeter long. There are 100 centimeters in a meter, and 1,000 meters in a kilometer. Estimate how many M&Ms it would take to measure the length of the Mississippi River
Answers
Length of Mississippi River = (4000 km)(1000 m/1 km)(100 cm/1 m)
= 400,000,000 cm
Since there are 400,000,000 cm in the entire length of the river, the answer would also be 400,000,000.
To measure the approximately 4,000 km long Mississippi River with M&Ms, we will need about 400,000,000 M&Ms, as there are 100 cm in a meter and 1,000 m in a km.
The question involves estimating the length of the Mississippi River using the size of M&Ms as a unit of measurement. Since the river is about 4,000 kilometers long and the standard unit of length in the SI system is the meter, we need to convert kilometers to centimeters to match the size of an M&M, which is about 1 centimeter long. There are 100 centimeters in a meter, and 1,000 meters in a kilometer. Therefore, we have:
Convert kilometers to meters: 4,000 km x 1,000 m/km = 4,000,000 m.Convert meters to centimeters: 4,000,000 m x 100 cm/m = 400,000,000 cm.Now, since each M&M is 1 centimeter, the number of M&Ms needed to measure the length of the Mississippi River is equal to the total number of centimeters:
400,000,000 M&Ms would be required to measure the approximate length of the Mississippi River.
For a normal distribution, the proportion in the tail beyond z = 0.30 is p = 0.1179.
a. True
b. False
Answers
Using the normal distribution, it is found that the statement is False, thus the correct option is b.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.The area on the tail beyond z is the proportion that is above z, that is, 1 subtracted by the p-value of z.For this problem, z = 0.3 has a p-value of 0.6554.
1 - 0.6554 = 0.3446
The proportion in the tail beyond z = 0.30 is p = 0.3446, thus, the statement is False.
A similar problem is given at https://brainly.com/question/12518252
A bus can carry 50 people per run. At least how many times does the bus have to run in order to transfer 25,000 people?
Answers
the slope of the line that passes through the points (-6,w) and (-10,4) is 1/8. what is the value of w
Answers
y-k = m(x-h), where (h,k) is any point on the line.
Suppose we first focus on the given point (-10,4). Let (h,k) = (-10,4).
Then y-k = m(x-h) becomes y-4 = m(x-[-10])
The other point is (-6,w), where w is the unknown we must determine.
w-4 = (1/8)(-6+10). Find w:
Mult both sides by 8 to remove the fraction 1/8:
8w-32 = -6 + 10 = 4. Then 8w = 32+4 = 36. w = 36/8 = 4.5, or 9/2 (ans.)
The value of 'w' using the given slope is [tex]\frac{9}{2}[/tex]
Given :
the slope of the line that passes through the points (-6,w) and (-10,4) is 1/8
apply slope formula
[tex]slope =\frac{y_2-y_1}{x_2-x_1}[/tex]
Substitute the values given in the points and make it equal to 1/8
[tex]slope =\frac{4-w}{-10+6}=\frac{1}{8} \\\frac{4-w}{-4}=\frac{1}{8}\\(4-w)(8)=-4(1)\\32-8w=-4\\-8w=-4-32\\-8=-36[/tex]
Divide both sides by -8
[tex]w=\frac{-36}{-8} =\frac{9}{2}[/tex]
The value of 'w' is [tex]\frac{9}{2}[/tex]
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Engineers are designing a box-shaped aquarium with a square bottom and an open top. The aquarium must hold 256 ft^3 of water. What dimensions should they use to create an acceptable aquarium with the least amount of glass?
Answers
Write an expression for the area of the aquarium's glass sides and bottom.
256 ft^3 256 ft^3
A= W^2 + 4(W*H) = W^2 + 4*W*(------------- ) = W^2 + 4*-------------------
W^2 W
So now we have A(W), the area as a function of W alone.
We want to minimize this area. To do this, differentiate A(W) with respect to W and set the result = to 0. We want to determine the "critical numbers."
dA/dW = 2W - 1024*W^(-2) = 0
1024
Then 2W = ---------------
W^2
2W^3 = 1024, so W^3 = 512, and W = third root of 512 = 8
If W = 8 ft, then L = 8 ft also. Since L*W*H = 256 ft^3,
L*W*H = 256 ft^3 = (8 ft)^2*H = 256 ft^3. Then H = 4
The acquarium dimensions are 8 by 8 by 4 feet. This keeps the area of the aquarium to a minimum.
To solve the problem we must know about the volume and the surface area of cuboids.
The length of the aquarium is 4, and the width of the aquarium is 8.
Given to us
The aquarium must hold 256 ft^3 of water.We know that the base of the aquarium should be a square, therefore, the length and the breadth of the aquarium will be the same.
Volume of the AquariumLength x breadth x height = 256 ft³
Length x Length x height = 256 ft³
Length ² x height = 256 ft³
[tex]L^2\times H = 256\\\\H = \dfrac{256}{L^2}[/tex]
Surface Area of the AquariumSurface Area of the Aquarium
= Area of the base + (4 x Area of the 4 sides)
[tex]= L^2 + 4(L\times H)[/tex]
Substituting the value of H,
[tex]= L^2 + 4L(\dfrac{256}{L^2})[/tex]
Length of the AquariumLet the surface area be a function of L, therefore,
[tex]f(L) = L^2 + 4L(\dfrac{256}{L^2})[/tex]
Differentiating the function to get the minimum,
[tex]f(L) = L^2 + 4L(\dfrac{256}{L^2})\\\\f(L) = L^2 + (\dfrac{1024}{L})\\\\f'(L) = 2L - (\dfrac{1024}{L^2})[/tex]
Equating against zero,
[tex]2L - (\dfrac{1024}{L^2}) = 0\\\\\dfrac{2L^3-1024}{L^2} = 0\\\\2L^3-1024 = 0\\\\L = 8[/tex]
Substituting the value of L,
[tex]H = \dfrac{256}{L^2}\\\\H = \dfrac{256}{8^2}\\\\H = 4[/tex]
But this way the area will be more, therefore, replace the value of L with 8,
L = 4
H = 8
Hence, the length of the aquarium is 4, and the width of the aquarium is 8.
Learn more about Volume of the cuboid:
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solve for x:
4/x-3 + 2/x^2-9 = 1/x+3
a. -17/3
b.13/3
c.19/3
d.-5
Answers
[tex]\dfrac{4(x+3)}{(x+3)(x-3)}+\dfrac2{x^2-9}=\dfrac{x-3}{(x+3)(x-3)}[/tex]
[tex]\dfrac{4x+12}{x^2-9}+\dfrac2{x^2-9}=\dfrac{x-3}{x^2-9}[/tex]
[tex]\dfrac{4x+12+2-(x-3)}{x^2-9}=0[/tex]
[tex]\dfrac{3x+17}{x^2-9}=0[/tex]
[tex]3x+17=0\implies x=-\dfrac{17}3[/tex]
The length of a rectangle is one unit shorter than one-sixth of the width, x. Which expression represents the perimeter of the rectangle? 7/3x−2 1/3x−4 1/3x−2 7/3x−8
Answers
Length: 1/6x-1
Perimeter
=x+x+1/6x+1/6x-1-1
=14/6x-2
=7/3x-2
Answer: [tex]\dfrac{7}{3}x-2[/tex]
Step-by-step explanation:
Let x be the width of the rectangle , then the length of the rectangle will be :_
Length = [tex]\dfrac{1}{6}x-1[/tex]
We know that the perimeter of a rectangle is given by :-
[tex]P=2(l+w)[/tex]
Substitute the value of length and width we get,
[tex]P=2(\dfrac{1}{6}x-1+x)[/tex]
Combine like terms, we get
[tex]P=2(\dfrac{1+6}{6}x-1)\\\\\Rightarrow\ P=2(\dfrac{7}{6}x-1)\\\\\Rightarrow\ P=2\times\dfrac{7}{6}x-2\times1\\\\\Rightarrow\ P=\dfrac{7}{3}x-2[/tex]
Hence, the expression represents the perimeter of the rectangle : [tex]\dfrac{7}{3}x-2[/tex]
He for got his pin code. 20% of his pin code was 246.8. What was his pin code?
Answers
So 246.8 * 5 = 1234
So his pin was 1234.
solve each inequality. Justify each step (y-) - 4 + 2y > 11
Answers
The perimeter of a rectangle is 42 cm. The longer side has length of 7 cm more than the shorter side. Find the length of longer side
Answers
2L+2W=42
Let's say the longer side is the length.
L=W+7
Let's plug that in for L.
2(w+7)+2w=42
2w+14+2w=42
Add like terms.
4w+14=42
Subtract 14 from both sides.
4w=28
Divide.
28÷4=7=W
The shorter side is 7, which means the longer side is 14 cm.
Which expression shows the result of applying the distributive property to 3(2x−6) ?
A. 2x−18
B. 6x−18
C. 6x−6
D. 5x−3
Answers
Answer:
b
Step-by-step explanation:
Determine the intercepts of the line. 7x-5=4y-6/7x−5=4y−6
Answers
Answer:
x.(-1/7,0)
y.(0,1/4)
Step-by-step explanation:
What expression is equivalent to x^2+2x-24/3x+18
Answers
The equivalent expression to [tex]x^2 + 2x - 24 / (3x + 18([/tex] is (x - 4) / 3, after factoring and simplifying the common terms.
The question is asking for an expression equivalent to [tex]x^2 + 2x - 24 / (3x + 18[/tex].
This expression can be factored and simplified.
First, notice that the quadratic expression in the numerator can be factored into (x + 6)(x - 4).
Similarly, the denominator can be factored as 3(x + 6). After factoring, we have:
= (x + 6)(x - 4) / 3(x + 6)
We can now simplify the expression by canceling out the common factor of (x + 6):
= (x + 6)(x - 4) / 3(x + 6) = (x - 4) / 3
So the equivalent expression is (x - 4) / 3.
192 beads to make a necklace and bracelet. 5 times as many beads to make a necklace than a bracelet. How many beads are used to make a the necklace?
Answers
A bicycle went 120 miles in 5 hours. A motorcycle is three times as fast as the bicycle. Find the speed of motorcycle.
Answers
Answer:
Step-by-step explanation:
120 x 5 = 600
The motorcycle is 3 times as fast.
600 x 3 = 1800
HELP!!! A pentagonal pyramid is sliced parallel to its base. what shape does it make?
Answers
same as the base
Good luck! =D
hich sentence is a run-on sentence?
A.
Dogs and hamsters are both fun pets, they relate to people in different ways.
B.
Dogs enjoy human contact; they love to fetch and chase.
C.
My hamster likes to run in his wheel, and he does not like to be held.
D.
When I try to pick up my hamster, he looks for a place to hide.
Answers
This question has been asked under the weong subject, but I'll answer anyway.
The correct answer, I believe, would be: A. Dogs and hamsters are both fun pets, they relate to people in different ways.
I really hope this helped you out! :)
Find the equation of the ellipse with the following properties. The ellipse with x-intercepts (2, 0) and (-2, 0); y-intercepts (0, 4) and (0, -4).
Answers
[tex]\bf \textit{ellipse, vertical major axis}\\\\ \cfrac{(x-{{ h}})^2}{{{ b}}^2}+\cfrac{(y-{{ k}})^2}{{{ a}}^2}=1 \qquad \begin{cases} center\ ({{ h}},{{ k}})\\ ------\\ h=0\\ k=0 \end{cases} \\\\\\ \cfrac{(x-0)^2}{2^2}+\cfrac{(y-0)^2}{4^2}=1\implies \cfrac{x^2}{4}+\cfrac{y^2}{16}=1[/tex]
write three different equivalent expressions for the following expression:
16x^8
Answers
Three different equivalent expressions for 16x^8 are [tex]\[ 16x^8 = 2^4 \cdot x^8 = (2 \cdot x)^4 \cdot x^4 = (2x)^4 \cdot x^4 \][/tex], [tex]\[ 16x^8 = 2^4 \cdot x^8 = (2x)^4 \][/tex], [tex]\[ 16x^8 = 2^4 \cdot x^8 = (2x)^4 \cdot x^4 = 2^{4 + 4} \cdot x^{4 + 4} = 2^8 \cdot x^8 \][/tex].
Expressing [tex]\(16x^8\)[/tex] in three different equivalent forms:
1. Expanded Form:
[tex]\[ 16x^8 = 2^4 \cdot x^8 = (2 \cdot x)^4 \cdot x^4 = (2x)^4 \cdot x^4 \][/tex]
2. Factored Form:
[tex]\[ 16x^8 = 2^4 \cdot x^8 = (2x)^4 \][/tex]
3. Exponential Form with a Common Base:
[tex]\[ 16x^8 = 2^4 \cdot x^8 = (2x)^4 \cdot x^4 = 2^{4 + 4} \cdot x^{4 + 4} = 2^8 \cdot x^8 \][/tex]
These expressions show the flexibility in representing the original expression using different algebraic manipulations. The key is to recognize patterns, use properties of exponents, and factor out common terms.
A person at ground level measures the angle of elevation to the top of a building to be 74 degrees. If at this point the person is 34 feet away from the base of the building, then how tall is the building? (Please round to the tenths place; please show work/steps and provide units with your answer) WILL GIVE BRAINIEST
Answers
to find the height of the building multiply the distance ( 34 feet) by the tangent of the angle(74)
34 * tan(74) = 118.57 feet
34 * tan(74) = 118.57 feet
Hope this helps
What happens to the arrangement of water molecules as ice melts?
A. Molecules gain enough energy to escape as vapor.
B. Molecules release enough energy to escape as vapor.
C. Molecules release enough energy to move from their fixed positions.
D. Molecules gain enough energy to move from their fixed positions.
Answers
suppose you deposit $25,000 in to a fund for which the annual interest rate is 5.3% compounded continuously. find the number of years it will take to double in value (round your answer to the nearest tenth of a year)
Answers
[tex]\bf \qquad \textit{Continuously Compounding Interest Earned Amount}\\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\to &\$50000\\ P=\textit{original amount deposited}\to& \$25000\\ r=rate\to 5.3\%\to \frac{5.3}{100}\to &0.053\\ t=years \end{cases} \\\\\\ 50000=25000e^{0.053t}\implies \cfrac{50000}{25000}=e^{0.053t}\implies 2=e^{0.053t} \\\\\\ ln(2)=ln(e^{0.053t})\implies ln(2)=0.053t\implies \cfrac{ln(2)}{0.053}=t \\\\\\ 13.0782486898102888\approx t\implies 13.08\approx t[/tex]
A recipe makes 6 2/3 cups the serving size is 5/6 cup how many servings does the recipe make
Answers
The recipe can make serving 8 cups.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
Here, given that,
A recipe makes 6 2/3 cups
=20/3 cups
And, the serving size is 5/6 cup
So, the recipe can make serving = 20/3÷ 5/6 cups
=20/3 × 6/5 cups
=8 cups
Hence, the recipe can make serving 8 cups.
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An athletic director pays $90 for 12 sun visors for the softball team. a. How much will the athletic director pay to buy 15 more sun visors?
Answers
90 / 12 = 7
$7.50 for a sun visor.
15 * 7.5 = 112.5
The athletic director needs to pay $112.5 to buy 15 more sun visors.
The temperature, t, in Burrtown starts at 32°F at midnight, when h = 0. For the next few hours, the temperature drops 2 degrees every hour.Which equation represents the temperature, t, at hour h?
Answers
Answer: We find the equation that represents the temperature (t) at hour (t) using the slope-intercept form and it is:
t = -2 h + 32
Solution:
The temperature, t, in Burrtown starts at 32°F at midnight, when h = 0. This represents the y-interception of the right line, when h=0 hours, t=32 °F:
Point=(h,t)→Point=(0,32)=(0,b)→b=32
For the next few hours, the temperature drops 2 degrees every hour. This represents the slope of the line (m): m=-2, negative because the temperature drops.
Slope-Intercept form:
y=mx+b; y=t, m=-2, x=h, b=32
Replacing the values:
t=-2h+32
Answer:
t=-2h+32
Step-by-step explanation:
An m-bit password is required to access a system. a hacker systematically works through all possible m-bit patterns. let x be the number of patterns tested until the correct password is found. find the conditional pmf of x given that the password has not been found after k tries
Answers
Final answer:
The conditional pmf of x, the number of patterns tested until the correct m-bit password is found, given that the password has not been found after k tries, is uniformly distributed with 1/(2^m - k) for x = k+1, k+2, ..., 2^m.
Explanation:
The question deals with the topic of probability and specifically centers around the concept of a conditional probability mass function (pmf). Given that an m-bit password is systematically being guessed by a hacker and has not been found after k attempts, we need to find the conditional pmf of x, where x is the number of patterns tested until the password is found.
Since the password has not been guessed within the first k tries, the conditional probability is based on the remaining 2m - k possibilities. The distribution of x will be uniform because each subsequent attempt has an equal probability of being correct. The conditional pmf of x given that the password was not found after k tries is 1/(2m - k) for x = k+1, k+2, ..., 2m.
What is the mean of the following set of numbers?
{-7, 23, -5, 4}
A. 3.75
B. 9.75
C. -9.75
D. -3.75
Answers
the mean is the average
-7+23 -5 +4 = 15
15 / 4 = 3.75
Answer is A
To calculate the mean, we add all the numbers together and divide the answer by how many numbers there are.
✦ Step 1: Add -7 + 23 + (-5) + 4
✦ = 15
✦ Step 2: Divide 15 by the total amount of numbers. There are four numbers, so divide 15 with four.
✦ = 3.75
Answer: Letter A ✅
Hope that helps! ★ If you have further questions about this question or need more help, feel free to comment below or post another question and send the link to me. -UnicornFudge aka Nadia
What are the solutions of 2x2 + 8x = −26
Answers
4+8x=-26
and 8x=-26-4
so 8x=-30
and we divide by 8 and x=-30/8=-15/4
Have fun
calculate the cost of fuel. 40 L at 65.7cents/L
Answers
Multiply 65.7 × 40 = 2,628
Then add the decimal between 26 and 28
What is cos θ if sec θ = 2?
Answers
Answer:
the value of cos θ is [tex]\frac{1}{2}[/tex].
Step-by-step explanation:
Since, [tex] cos\theta=\frac{1}{sec\theta}[/tex]
we are given with sec θ = 2
To find the value of cos θ , we use this fact [tex] cos\theta=\frac{1}{sec\theta}[/tex]
put sec θ = 2 in [tex] cos\theta=\frac{1}{sec\theta}[/tex]
[tex] cos\theta=\frac{1}{2}[/tex]
Therefore, the value of cos θ is [tex]\frac{1}{2}[/tex].