this is super confusing
A cardboard box without a top is to have volume 62500 cubic cm. find the dimensions which minimize the amount of material used.
The dimensions of the box that minimize the amount of material used are approximately 39 cm by 39 cm by 39 cm.
To minimize the amount of material used, we need to minimize the surface area of the box.
Let's denote the dimensions of the box as x,y, and z, where x and y are the dimensions of the base (length and width), and z is the height.
The volume V of the box is given by:
[tex]\[ V = xyz \][/tex]
We're given that V = 62500 cubic cm.
The surface area A of the box (excluding the top) is given by:
[tex]\[ A = xy + 2xz + 2yz \][/tex]
To minimize the amount of material used, we need to minimize A subject to the constraint V = 62500 .
We can solve this problem using the method of Lagrange multipliers. First, let's define the Lagrangian function L as follows:
[tex]\[ L(x, y, z, \lambda) = xy + 2xz + 2yz + \lambda(62500 - xyz) \][/tex]
Now, we take partial derivatives of L with respect to x,y,z, and [tex]\( \lambda \),[/tex] and set them equal to zero:
[tex]\[ \frac{\partial L}{\partial x} = y + 2z - \lambda yz = 0 \]\[ \frac{\partial L}{\partial y} = x + 2z - \lambda xz = 0 \]\[ \frac{\partial L}{\partial z} = 2x + 2y - \lambda xy = 0 \]\[ \frac{\partial L}{\partial \lambda} = 62500 - xyz = 0 \][/tex]
Solving these equations will give us the dimensions that minimize the amount of material used.
However, this system of equations is quite complex to solve manually. Let's simplify the problem by recognizing that the dimensions that minimize the amount of material used will likely be those where the box is as close to a cube as possible. This means x=y to minimize the perimeter, and x=y=z to minimize the surface area.
So, let's find the cube root of 62500 :
[tex]\[ \sqrt[3]{62500} \approx 38.99 \][/tex]
Since [tex]\( 38^3 = 54872 \)[/tex] and [tex]\( 39^3 = 59319 \)[/tex], the closest perfect cube to 62500 is [tex]\( 39^3 \)[/tex]. Therefore, the dimensions of the box that minimize the amount of material used are approximately 39 cm by 39 cm by 39 cm.
Find the area of a parallelogram that has a base of 24 feet and a height of 12 feet
you're answer would be two hundred eighty eight
Answer:
288
Explanation:
Hope this helps
For every pizza Eric's family ate, eric ate2 of the 8 pieces. If Eric's family bought 2 pizzas, write the ratio of the total number of pieces Eric ateto the total number of pieces the family ate.
Answer:
4 over sixteen, or 4/16 is your answer.
Step-by-step explanation:
There are 2 pizzas, each of which contain 8 pieces. So, in all, there were 8 pieces. Eric ate two pieces from each pizza. So, 2+2=4. Thus, he ate four out of the 16 pieces. That brings us to our answer, 4/16.
I hope this helps you, have a wonderful day!
Final answer:
The ratio of the total number of pieces Eric ate to the total number of pieces eaten by his family when they bought 2 pizzas, is 1:4, meaning Eric eats 1 out of every 4 pieces.
Explanation:
For every pizza Eric's family ate, Eric ate 2 of the 8 pieces. If Eric's family bought 2 pizzas, to find the ratio of the total number of pieces Eric ate to the total number of pieces the family ate, we start by calculating the total pieces in 2 pizzas and the pieces Eric ate.
Each pizza has 8 pieces, so 2 pizzas have 8 * 2 = 16 pieces in total.
Since Eric eats 2 pieces per pizza, for 2 pizzas, he would eat 2 * 2 = 4 pieces in total.
Therefore, the ratio of the pieces Eric ate to the total pieces is:
4 (pieces Eric ate) : 16 (total pieces), which simplifies to 1:4.
This ratio means that for every piece of pizza Eric eats, there are 4 pieces total, indicating Eric eats 1 out of every 4 pieces.
g A university administrator obtains a sample of the academic records of past and present scholarship athletes at the university. The administrator reports that no significant difference was found in the mean GPA (grade point average) for male and female scholarship athletes (P = 0.287). This conclusion implies that the maximum difference in GPAs between male and female scholarship athletes is 0.287. the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample if there is no difference in mean GPAs is 0.287. the GPAs for male and female scholarship athletes are identical, except for 28.7% of the athletes. the chance that a pair of randomly chosen male and female scholarship athletes would have a significant difference in GPAs is 0.287
Answer:
the chance of obtaining a difference in GPAs between male and female scholarship athletes as large as that observed in the sample if there is no difference in mean GPAs is 0.287
Step-by-step explanation:
When conducting hypothesis tests, the P value tells you the probability of the results happening by random chance. That's why we usually test against a very low alpha level, usually 1% or 5%.
Here P = 28%, so there is a 28% chance that if we take a sample of athletes, about 1 out of 4 times there won't be a significant difference in GPA between males and females.
Please help me if you can!
Here is your answer
[tex]<b>z= 20 degrees</b>[/tex]
REASON:
[tex]<font color="blue" size=5>Concept used</font>[/tex]: The sum of adjacent angles of a parallelogram is 180 degrees.
So, in above given figure
[tex] 2z+16+124=180 [/tex] (measures of adjacent angles)
[tex]2z+140=180 [/tex]
[tex] 2z=180-140 [/tex]
[tex]2z=40 [/tex]
[tex]z= 40/2 [/tex]
[tex]z= 20 [/tex]
HOPE IT IS USEFUL
Find two positive even consecutive integers such that the square of the smaller integer is 10 more than the larger integer
Answer:
the numbers are [tex]4,6[/tex]
Step-by-step explanation:
Let
x-------> the smaller even consecutive integer
x+2-------> the larger even consecutive integer
we know that
[tex]x^{2}=(x+2)+10[/tex]
solve for x
[tex]x^{2}-x-12=0[/tex]
The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]x^{2}-x-12=0[/tex]
so
[tex]a=1\\b=-1\\c=-12[/tex]
substitute in the formula
[tex]x=\frac{1(+/-)\sqrt{-1^{2}-4(1)(-12)}} {2(1)}[/tex]
[tex]x=\frac{1(+/-)\sqrt{49}} {2}[/tex]
[tex]x=\frac{1(+/-)7} {2}[/tex]
[tex]x=\frac{1(+)7} {2}=4[/tex] ------> the solution (must be positive)
[tex]x=\frac{1(-)7} {2}=-3[/tex]
therefore
the numbers are [tex]4,6[/tex]
Two positive even consecutive numbers are [tex]4,6[/tex].
Let [tex]x,\;x+2[/tex] are two positive consecutive even number.
According to question,
[tex]x^2=(x+2)+10[/tex]
[tex]x^2-x-12=0[/tex]
Solve the quadratic equation,
[tex]x^2-4x+3x-12=0\\x(x-4)-3(x-4)=0\\(x-4)(x-3)=0\\\; x-4=0\\ \; x-3=0\\[/tex]
So [tex]x=3 , 4[/tex].
Positive even number is the requirement so [tex]3[/tex] is not a even number so eliminate.
Hence value of [tex]x[/tex] is [tex]4[/tex].
Two positive even consecutive numbers are [tex]4,6[/tex].
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Given that the points (-5,7), (5,7),(5,1), and (-5,1) are vertices of a rectangle, how much stronger is the width than the length
Answer:
4 units
Step-by-step explanation:
The sides of the rectangle are parallel to the axes, so it is a simple matter to subtract coordinate values to find the dimensions.
Along the line y=7, the rectangle extends from -5 to 5, so has width 10.
Along the line x=5, the rectangle extends from 1 to 7, so has length 6.
The width is 10 - 6 = 4 units more than the length.
Answer: width=7 length =9
9-7=2
2 is the answer
What is the m∠N ? Can someone help me :)
Answer:
61°Step-by-step explanation:
Use the sine law:
[tex]\dfrac{MO}{\sin(\angle N)}=\dfrac{NM}{\sin(\angle O)}[/tex]
We have:
[tex]MO=18\\\\NM=6\\\\m\angle O=17^o\to\sin17^o\approx0.2924[/tex]
Substitute:
[tex]\dfrac{18}{\sin(\angle N)}=\dfrac{6}{0.2924}[/tex] cross multiply
[tex]6\sin(\angle N)=(18)(0.2924)[/tex]
[tex]6\sin(\angle N)=5.2632[/tex] divide both sides by 6
[tex]\sin(\angle N)=0.8772\to m\angle N\approx61^o[/tex]
A pyramid has a cross-sectional shapes, taken parallel to its base, that are _____ to one another
Answer: similar
Step-by-step explanation: apex
Answer:
The fill in the blank is the word similar.
Step-by-step explanation:
A pyramid has a cross-sectional shapes, taken parallel to its base, that are similar to one another.
When we do the cross-section of a pyramid parallel to its base, the cross-section formed will be in the same shape like the base.
Whenever a cross section is done parallel to the base of any pyramid, the resulting shape will be same like the base. As we move up in the pyramid and cross section it, the shape formed will be like the base but with smaller dimensions. But the shape will be the same as the base.
A party store sells large plates in packs of 12 and small plates in packs of eight. In order to have an equal number of both, what is the least amount of large plaate packs that would have to be purchased?
Answer:
The least amount of large plate packs that would have to be purchased is 24.
Step-by-step explanation:
To do this we have to find first multiplication of 8 that is equally divided by 12.
A person can buy large packs of plates in multiple of 12 only.
A person can buy {1,2,3,4,5,6,7,8,...}packs of plates and will purchase
{12,24,36 ,48,60,72,84,96,...}large plates.
A person can buy small packs of plates in multiple of 8 only.
A person can buy {1,2,3,4,5,6,7,8,9,10,11,12...}packs of plates and will purchase..
{8,16,24,32,40,48,56,64,72,80,88,96,...}small plates
Now we have to keep in mind that one has to purchase equal amount of plates..
A person can buy {24,48,72,96} plates and hence least amount of large plate packs that would have to be purchased is 24....
A sample in which every person object or event has a equal chance of being selected
Answer:
Random Sample
Step-by-step explanation:
Three students are working to find the solution set of this system of equations: y = 3x + 10 2y = 6x – 4
Answer:
hope this helps!
Step-by-step explanation:
The solution set of the given system of equations y = 3x + 10 and 2y = 6x – 4 is empty, indicating that the lines are parallel and do not intersect.
Explanation:To find the solution set of the system of equations y = 3x + 10 and 2y = 6x – 4, we can solve the equations simultaneously by substitution or elimination method.
Using the substitution method:
Step 1: Substitute the value of y from the first equation into the second equation. 2(3x + 10) = 6x – 4.Step 2: Simplify the equation: 6x + 20 = 6x – 4.Step 3: Combine like terms and isolate the variable: 6x - 6x = -4 - 20.Step 4: Simplify further: 0 = -24.Step 5: Since 0 does not equal -24, the system of equations does not have a solution.Therefore, the solution set of the system of equations y = 3x + 10 and 2y = 6x – 4 is empty, indicating that the lines represented by the equations do not intersect and are parallel.
Solve: 3x(x + 4) + 3(x + 4) = 0 A) 0 and -4 B) -1 and 4 C) -1 and -4 Eliminate D) 1 and 4
Answer:
The answer is C) -1 and -4
Step-by-step explanation:
Sorry i don't have the step by step explanation but that's the answer
Find two numbers whose difference is 102 and whose product is a minimum. Step 1 If two numbers have a difference of 102, and one of them is x + 102, then the other is $$ Incorrect: Your answer is incorrect. x. Step 2 The product of two numbers x and x + 102 can be simplified to be x2 Correct: Your answer is correct. seenKey 2 + 102 Correct: Your answer is correct. seenKey 102 x. Step 3 If f(x) = x2 + 102x, then f '(x) = $$ Correct: Your answer is correct. 2x+102. Step 4 To minimize the product f(x) = x2 + 102x, we must solve 0 = f '(x) = 2x + 102, which means x = -51 Correct: Your answer is correct. seenKey -51 . Step 5 Since f ''(x) = 2 , there must be an absolute minimum at x = −51. Thus, the two numbers are as follows. (smaller number) (larger number)
Answer:
The two numbers would be -51 and 51
Step-by-step explanation:
To find these, first set the equation for the first number as x. You can then set the second number as x + 102. Now, find their product.
x(x + 102) = x^2 + 102x
Now, to find the minimum, find the value of x in the vertex of this equation.
-b/2a = -102/2(1) = -102/2 = -51
So we know -51 is the first number. Now we find the second using the prewritten equation.
x + 102 = -51 + 102 = 51
The two numbers whose difference is 102 and whose product is a minimum are -51 and 51. This is obtained by differentiating and finding the minimum of the function that represents their product.
Explanation:To find two numbers whose difference is 102 and whose product is a minimum, we initially express the two numbers as x and x + 102. The product of these two numbers can be denoted as f(x) = x(x + 102) = x² + 102x. To find the minimum product, we differentiate f(x) to find f'(x) = 2x + 102. Setting this equal to zero gives x = -51. We should note, f''(x) = 2, verifies that there is a minimum at x = -51. Hence, the two numbers are -51 and (-51 + 102) = 51.
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Bethany sells roses and petunias. The expression 3r+2.5p3r+2.5p3, r, plus, 2, point, 5, p gives the cost (in dollars) of r roses and p petunias. What is the cost of 7 roses and 8 petunias?
Lisa purchased groceries worth $34 for a party. A gallon of milk costs $2.50, a gallon of ice cream costs $7.50, and a gallon of lemonade costs $3.00
If Lisa bought 4 gallons of milk, and 2 gallons of ice cream, how many gallons of lemonade did she buy?
Answer 1: 5
Answer 2: 3
Answer 3: 2
Answer 4: 4
Answer 2 she bought 3 gallons of lemonade
4x2.50=10
2x7.50=15
10+15=25
34-25=9
9 divided by 3=3
Answer:
Answer 2 is right.
Step-by-step explanation:
Given that Lisa purchased groceries worth $34 for a party.
A gallon of milk costs $2.50
No of gallons bought - 4
Cost of milk = [tex]2.50*4 =10[/tex]
a gallon of ice cream costs $7.50
Cost of ice creams = [tex]2*7.50 = 15[/tex]
, and a gallon of lemonade costs $3.00
Let lemonade purchased be x
Then total cost = [tex]10+15+3x[/tex]
This equals 34
[tex]25+3x =34\\3x=9\\x=3[/tex]
No of gallons of lemonade purchased =3
Answer 2 is right
You deposit $300 in an account earning 5% interest compound annually. How much will you have in the account after 10 years
Answer:
450
Step-by-step explanation:
5% of 300 = 15
300+10(15)=450
Using the compound interest formula, if you deposit $300 in an account earning 5% interest compounded annually, you would have approximately $488.85 in the account after 10 years.
Explanation:This problem involves compound interest, which appeal to financial maths. The formula for compound interest is A = P*(1 + r/n)^(nt), where:
A is the amount of money accumulated after n years, including interest.P is the principal amount (the initial amount of money).r is the annual interest rate (in decimal).n is the number of times that interest is compounded per year.t is the time the money is invested for, in years.For this problem, P = $300, r = 5% or 0.05, n = 1 (because it's compounded annually), and t = 10 years. Substituting these values into the formula, you have A = 300*(1 + 0.05/1)^(1*10).
Doing the math, A = $300 * (1.05)^10 ≈ $488.85. So after 10 years, you would have approximately $488.85 in the account.
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A 5.5-foot-tall woman walks at 4 ft/s toward a street light that is 27.5 ft above the ground. what is the rate of change of the length of her shadow when she is 16 ft from the street light? at what rate is the tip of her shadow moving?
The scenario involves differentiating a proportional relationship between the woman's distance from a streetlight and the length of her shadow to determine the rate of change of the shadow's length and the speed of the shadow's tip.
Explanation:The question involves finding the rates of change in the scenario where a woman walks toward a streetlight and the effects on her shadow's length and speed. Let's denote the distance between the woman and the streetlight as x, the length of her shadow as y, and her height as h. The streetlight's height is given as H. Using similar triangles, we can establish the relationship (H - h) / y = H / (x + y). Differentiating both sides with respect to time, t, allows us to calculate the rates of change we're interested in.
To find the rate of change of her shadow's length when she is 16 feet from the streetlight, we can take dh/dt as 0 since her height is constant, dx/dt as -4 ft/s because she is walking towards the light and substitute into the derived equation after differentiating. Similarly, we calculate the rate at which the tip of the shadow is moving by adding the rate of the woman's movement to the rate of change of the shadow's length.
True or false (Picture provided)
The answer is: True.
Why?A counterexample is a way that we can prove that something is not true about a mathematical equation or expression, it's also considered as an exception to a rule.
So:
[tex]sec^{2}x-1=\frac{cosx}{cscx}\\\\tg^{2}x=\frac{cosx}{\frac{1}{sinx}}\\\\\frac{1-cos2x}{1+cos2x}=cosxsinx[/tex]
Then, evaluating we have:
[tex]\frac{1-cos(2*45)}{1+cos(2*45)}=cos(45)*sin(45)\\\\\frac{1-0}{1+0}=\frac{\sqrt{2} }{2}*\frac{\sqrt{2}}{2}\\\\1=\frac{(\sqrt{2})^{2} }{4}\\\\1=\frac{2}{4}\\\\1=\frac{1}{2}[/tex]
Hence, we can see that the equation is not fulfilled, so, 45° is a counterexample for [tex]sec^{2}x-1=\frac{cosx}{cscx}[/tex] and the answer is true.
Have a nice day!
Let f(x)=100 / −10+e^−0.1x .
What is f(−11) ?
Enter your answer, rounded to the nearest tenth, in the box
Answer:
-14.29422
Step-by-step explanation:
rounded to the nearest tenth = -14.3
What is measure of angle A?
Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth.
Answer:
A = 53.13 degrees
Step-by-step explanation:
sin theta = opposite side/ hypotenuse
sin A = 4/5
Take the inverse of each side
sin ^-1 sin A = sin ^-1 (4/5)
A = 53.13010
To the nearest hundredth
A = 53.13 degrees
David sold 6 apple trees. He sold 5 of the apple trees to max. What fraction of the apple trees did david sell to max
Answer:
wouldn't it be 6/5 or 5/6 or 1.2
Step-by-step explanation:
Describe the translation of f(x) = |x|.
4 1\2 left, 1\2 unit down
1\2 unit left, 4 1\2 up
1\2unit left, 4 1\2units down
4 1\2 units left, 1\2 unit up
Answer:
the correct answer would be choice C
it went left 1/2 units and went down 4 1/2 units
Answer:
It's the third option.
Step-by-step explanation:
f(x) = |x| is shaped like a V with the vertex at the point (0, 0).
The translation is 1/2 unit to the left, 4 1/2 units down.
If you deposit $300 in an account with a 6% interest rate, how much will be in your account after 1 year?
assuming simple interest rate.
[tex]\bf ~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$300\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ t=years\dotfill &1 \end{cases} \\\\\\ A=300[1+(0.06)(1)]\implies A=300(1.06)\implies A=318[/tex]
Answer:
$318
Step-by-step explanation:
I was just in a zoom class that used this as an example
what is the value of the constant in the equation that relates the height and width of this rectangle?
would appreciate any help i could get :)
Answer:
Option C. [tex]2.5[/tex]
Step-by-step explanation:
we know that
In the rectangle of the figure
[tex]\frac{H}{W}=\frac{25}{10}=2.5[/tex]
so
[tex]H(W)=2.5W[/tex]
The constant is equal to [tex]2.5[/tex]
Answer:
2.5
Step-by-step explanation:
A jar contains 38 marbles. It has 10 red, 22 black and 6 green marbles. Two marbles are drawn, the first is not returned before the second one is drawn. What is the probability that both marbles are black?
P(Both Black) = 11 / 19
P(Both Black) = 231 / 703
P(Both Black) = 231 / 722
P(Both Black) = 121 / 361
Answer:
The correct answer option is 231 / 703
Step-by-step explanation:
We are given that a jar has 38 marbles, out of which 10 are red, 22 are black and 6 are green. Two marbles are drawn and the first marble is not returned when the second one is drawn.
We are to find the probability that both marbles are black.
1st draw: P (black) = [tex]\frac{22}{38} =\frac{11}{19}[/tex]
2nd draw: P (black) = [tex]\frac{21}{37} [/tex]
P (Both Black) = [tex]\frac{11}{19} \times \frac{21}{37}[/tex] = 231 / 703
Solve the equation (linear equation)
[tex]8^{2x+7} = (\frac{1}{32})^{3x}[/tex]
Answer: [tex]x=-1[/tex]
Step-by-step explanation:
By the negative exponent rule, you have that:
[tex](\frac{1}{a})^n=a^{-n}[/tex]
By the exponents properties, you know that:
[tex](m^n)^l=m^{(nl)}[/tex]
Therefore, you can rewrite the left side of the equation has following:
[tex](\frac{1}{8})^{-(2x+7)}=(\frac{1}{32})^{3x}[/tex]
Descompose 32 and 8 into its prime factors:
[tex]32=2*2*2*2*2=2^5\\8=2*2*2=2^3[/tex]
Rewrite:
[tex](\frac{1}{2^3})^{-(2x+7)}=(\frac{1}{2^5})^{3x}[/tex]
Then:
[tex](\frac{1}{2})^{-3(2x+7)}=(\frac{1}{2})^{5(3x)}[/tex]
As the base are equal, then:
[tex]-3(2x+7)=5(3x)[/tex]
Solve for x:
[tex]-6x-21=15x\\-21=15x+6x\\-21=21x\\x=-1[/tex]
How can a graph help you determine an exponential model is appropriate for a data set? Explain
The graph of the data set will help you to understand if the data is displayed in the form of an exponential model. Example: if the data (when graphed) is similar in shape and growth to that of the graph of f(x)=e^x, then it could be deemed helpful.
Ivan wants to ship a box that is 6 in. Tall, 4 in. Wide, and 5 in. Long to his cousin, and used the calculation below. What was his error, and what is the correct volume?
Answer:
sample response He added instead of multiplying 20 and 6. The correct answer is 120 cubic inches.
Answer:
He added instead of multiplying 20 and 6. The correct answer is 120 cubic inches.
That is the correct answer
Step-by-step explanation:
Let p(x)=90/9+50e^-x What is p(3)
Answer:
If it is [tex]p(x)=\frac{90}{(9+50e^{-x})}[/tex], then: [tex]p(3)=7.83[/tex]
If it is [tex]p(x)=\frac{90}{9}+50e^{-x}[/tex], then: [tex]p(3)=12.48[/tex]
Step-by-step explanation:
To solve this exercise you must substiute x=3 into the expression given in the problem.
1) If the expression is [tex]p(x)=\frac{90}{(9+50e^{-x})}[/tex], you obtain:
[tex]p(3)=\frac{90}{(9+50e^{-(3)})}[/tex]
[tex]p(3)=7.83[/tex]
2) If the expression is [tex]p(x)=\frac{90}{9}+50e^{-x}[/tex], you obtain:
[tex]p(3)=\frac{90}{9}+50e^{-(3)}\\p(3)=12.48[/tex]
Answer:
p (3) = 7.83
Step-by-step explanation:
We are given the following expression and we are to evaluate it given that the value of [tex] p = 3 [/tex]:
[tex] p ( x ) = \frac { 9 0 } { 9 + 5 0 e^ { - x } } [/tex]
Substituting the given value of [tex] p [/tex] to get:
[tex] p ( 3 ) = \frac { 9 0 } { 9 + 5 0 e^ { - 3 } } [/tex]
[tex] p ( 3 ) = \frac { 9 0 } { 11.489 } [/tex]
[tex] p ( 3 ) = 7.83 [/tex]