Mathematics College

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 181.5 square centimeters

Answers

Answer 1

Answer:

The base and height of the solid is 5.5cm

Step-by-step explanation:

Given

[tex]Surface\ Area = 181.5cm^2[/tex]

Required

Determine the dimensions that maximizes the volume

Let the base dimension be x and the height be h

The volume is calculated as:

[tex]Volume =x^2 * h[/tex]

[tex]Volume =x^2h[/tex]

181.5 =x^2h

The surface area (S) is calculated as this:

[tex]S = 2(x^2 + xh + xh)[/tex]

[tex]S = 2(x^2 + 2xh)[/tex]

[tex]S = 2x^2 + 4xh[/tex]

Substitute 181.5 for S

[tex]181.5 = 2x^2 + 4xh[/tex]

Make h the subject:

[tex]4xh = 181.5 - 2x^2[/tex]

[tex]h = \frac{181.5 - 2x^2}{4x}[/tex]

Substitute [tex]h = \frac{181.5 - 2x^2}{4x}[/tex] in [tex]Volume =x^2h[/tex]

[tex]V = x^2(\frac{181.5 - 2x^2}{4x})[/tex]

[tex]V = x(\frac{181.5 - 2x^2}{4})[/tex]

[tex]V = \frac{1}{4}(x)(181.5 - 2x^2)[/tex]

[tex]V = \frac{181.5x}{4} - \frac{2x^3}{4}[/tex]

[tex]V = \frac{181.5x}{4} - \frac{x^3}{2}[/tex]

To get the maximum, we differentiate V with respect to t and set the differentiation to 0

[tex]dV = \frac{181.5}{4} - \frac{3x^2}{2}[/tex]

Set to 0

[tex]0 = \frac{181.5}{4} - \frac{3x^2}{2}[/tex]

[tex]\frac{3x^2}{2} = \frac{181.5}{4}[/tex]

Multiply through by 4

[tex]4 * \frac{3x^2}{2} = \frac{181.5}{4} * 4[/tex]

[tex]2*3x^2 = 181.5[/tex]

[tex]6x^2 = 181.5[/tex]

[tex]x^2 = \frac{181.5}{6}[/tex]

[tex]x^2 = 30.25[/tex]

[tex]x = \sqrt{30.25[/tex]

[tex]x = 5.5[/tex]

Recall that:

[tex]h = \frac{181.5 - 2x^2}{4x}[/tex]

[tex]h = \frac{181.5 - 2 * 5.5^2}{4 * 5.5}[/tex]

[tex]h = \frac{181.5 - 60.5}{22}[/tex]

[tex]h = \frac{121}{22}[/tex]

[tex]h = 5.5[/tex]

So, we have:

[tex]h = 5.5[/tex]

[tex]x = 5.5[/tex]

Hence, the base and height of the solid is 5.5cm

Related Questions

please help thx so much

Answers

Answer:

I think c is the answer....

What is the unit rate of 15 laps in 3 days

Answers

The answer is going to be 5 laps per day because I did 15 laps/3days and divided each side by 3 which the answer is going to be 5/1 but it’s basically 5 since it’s going to be a whole number in this set. Therefore the unit rate of 15 laps in 3 days is going to be 5 laps per day.

Answer:

the answer to that is 5 laps per day

Step-by-step explanation:

15 laps ÷ 3

3 days ÷3

5 laps

1 day

5 laps=days

Jen has 81 beads. Ana has 44 red beads and 29 blue beads.
How many more beads does Ana need to have 81 beads also?
Ana needs
more beads to have 81 beads.

Answers

You have to add the 44 and 29

44+29 = 73 <— that’s the total number of beads Ana has. If she wants 81 we will have to use subtraction now. 81-73 = 8

So Ana will need 8 more beads in order to have 81 total beads

can somebody pls help me..

Answers

If you solve for the vertex of the function, you would get (-3, -1). So I think it would be x = -3 and the function’s maximum value would be -1. Hope this helps!! :)

Question 2
2 pts
Use the midpoint formula to find the midpoint of the line segment AB. Show your work for
full credit.
Given (4,8) and B(14,-2)
HTML Editore

Answers

Answer:

[tex]M = (9,3)[/tex]

Step-by-step explanation:

Given

[tex](x_1,y_1) = (4,8)[/tex]

[tex](x_2,y_2) = (14,-2)[/tex]

Required

Find the midpoint (M)

The midpoint is calculated as follows:

[tex]M = (\frac{1}{2}(x_1+x_2),\frac{1}{2}(y_1+y_2))[/tex]

Substitute values for x's and y's

[tex]M = (\frac{1}{2}(4+14),\frac{1}{2}(8-2))[/tex]

[tex]M = (\frac{1}{2}(18),\frac{1}{2}(6))[/tex]

[tex]M = (\frac{1}{2}*18,\frac{1}{2}*6)[/tex]

[tex]M = (9,3)[/tex]

Hence, the midpoint is (9,3)

how do I find the range of a function?

Answers

Answer:

Overall, the steps for algebraically finding the range of a function are:

1.Write down y=f(x) and then solve the equation for x, giving something of the form x=g(y).

2.Find the domain of g(y), and this will be the range of f(x).

3.If you can't seem to solve for x, then try graphing the function to find the range.

Step-by-step explanation:

Answer:

Write down y=f(x) and then solve the equation for x, giving something of the form x=g(y).

Find the domain of g(y), and this will be the range of f(x).

If you can't seem to solve for x, then try graphing the function to find the range.

Step-by-step explanation:

Which two lines are parallel?
5y = 2x-5
5y = 4 +3x
5y-3x = -1
O No two of the lines are parallel.
O I and III
O II and III
O land I

Answers

Given:

The equations are

[tex]5y=2x-5[/tex]

[tex]5y=4+3x[/tex]

[tex]5y-3x=-1[/tex]

To find:

The two parallel lines.

Solution:

Slope intercept form of a line is

[tex]y=mx+b[/tex]

where, m is slope and b is y-intercept.

Isolate variable y in each of the given equations, to get the slope intercept form of the lines.

The slope intercept forms of given lines are

[tex]y=\dfrac{2}{5}x-1[/tex]

[tex]y=\dfrac{4}{5}+\dfrac{3}{5}x[/tex]

[tex]y=-\dfrac{1}{5}+\dfrac{3}{5}x[/tex]

On comparing these equations with slope intercept form, we get

Slope of I line = [tex]\dfrac{2}{5}[/tex]

Slope of II line = [tex]\dfrac{3}{5}[/tex]

Slope of III line = [tex]\dfrac{3}{5}[/tex]

Slope of parallel lines are equal. So, lines II and III are parallel because their slopes are equal.

Therefore, the correct option is C.

Which route is longer, Kingston- Edmonds or Sourworth-Fauntleroy?

Answers

Kingston Edmonds
It’s longer, also I think I did this before

need answer for number 2

Answers

Answer:

9x^6 over 4y^8

Step-by-step explanation:

Answer:

E. 9x^6 / 4y^8

Step-by-step explanation:

Use exponent rule: a^-b = 1/a^b

What input is NOT allowed for f(x) = 3/ x-5

Answers

Answer:

x ≠ 5

Step-by-step explanation:

x ≠ 5

The reason for this is because if we use the input 5, this will be the result:

f(x) = 3/(5) - 5

f(x) = 3/0 = Undefined

PLEASEEE HELP!!! Need help pls

Answers

Answer:

Option A

Step-by-step explanation:

Weight of an apple = twice of the weight of a lemon

Weight of an apple = 2(weight of a lemon)

x = 2y ------(1)

x = weight of an apple

y = weight of a lemon

Number of apples in a box = 16

Weight of the apple box (A) = 16x

Number of lemons in another box = 24

Weight of the lemon box (B) = 24y

Therefore, [tex]\frac{A}{B}=\frac{16x}{24y}[/tex]

[tex]\frac{A}{B}=\frac{2x}{3y}[/tex]

[tex]\frac{A}{B}=\frac{2(2y)}{3y}[/tex] [From equation (1)]

[tex]\frac{A}{B}=\frac{4y}{3y}[/tex]

[tex]\frac{A}{B}=\frac{4}{3}[/tex]

Option A will be the correct option.

One professor grades homework by randomly choosing 5 out of 14 homework problems to grade.
(a) How many different groups of 5 problems can be chosen from the 14 problems?
(b) Probability extension: Jerry did only 5 problems of one assignment. What is the probability that the problems he did comprised the group that was selected to be graded? (Round your answer to four decimal places.)
(c) Silvia did 7 problems. How many different groups of 5 did she complete? What is the probability that one of the groups of 5 she completed comprised the group selected to be graded? (Round your answer to four decimal places.)

Answers

Answer:

a) 2002 different groups of 5 problems can be chosen from the 14 problems.

b) P=0.04995%

c) 21 different groups of 5.

P=1.04895%

Step-by-step explanation:

In order to solve this problem we must ask ourselves: Does the order matter? We can see that in this case the order in which the problems are chosen doesn't really matter, so we are talking about a combination:

[tex]_{n}C_{r}=\frac{n!}{r!(n-r)!}[/tex]

in this case, n=14 while r=5, so our combination will be:

[tex]_{14}C_{5}=\frac{14!}{5!(14-5)!}[/tex]

which yields:

[tex]_{14}C_{5}=2002[/tex] which will be the answer for part a.

In order to find the probability we can use the probability formula:

[tex]Probability=\frac{desired \hspace{5}outcome}{number\hspace{5} of \hspace{5} possible \hspace{5} outcomes}[/tex]

so in this case there is only 1 desired outcome out of the 2002 possible outcomes so our probability is:

[tex]P=\frac{1}{2002}[/tex]

which is the same as:

P=0.04995%

c) The same applies to part c. In order to find the number of groups Silvia did, we can make us of the combination formula:

[tex]_{7}C_{5}=\frac{7!}{5!(7-5)!}[/tex]

which yields:

[tex]_{14}C_{5}=21[/tex]

so there are 21 different groups of 5 she did by solving 7 problems.

So we can use the same formula to find the probability that one of the groups she completed comprised the group selected to be graded so we get:

[tex]P=\frac{21}{2002}=\frac{3}{286}[/tex]

which is the same as:

P=1.04895%

(remember that in order to turn a decimal number into a percentage you need to multiply the decimal number by 100)

Solve For X
5(3x-2)=5

Answers

Answer:

15x-10=5

x=1

Step-by-step explanation:

hope it helps bro

Answer:

x=1

Step-by-step explanation:

Please determine x.

Answers

Answer:

x = 15

Step-by-step explanation:

137+28=165

All angles in a triangle add up to 180

So 180-165 = 15

15 = x

Question 5 6 pts Alanis goes to work with her father. He gives out the paychecks to the employees at his company. Alanis notices that one man worked 31.25 hours for the week, and he was paid $9.20 per hour. How much was the man's paycheck? O A. $270.00 O B. $287.50 c. $1 150.00 O D. $1,437.50 Question 6 6 pts

Answers

The answer for your question is

Complementary angles are 2 angles with a sum of 90.

Answers

Answer:

Two angles are called complementary when their measures add to 90 degrees. Two angles are called supplementary when their measures add up to 180 degrees.

the table represents a function:

x. f(x)
-5. 4
-1. 0
6. -1
9. -3

What is the value of f(-1)?

Answers

Answer:

➨ f(-1) = 0

Step-by-step explanation:

➨ Look at the table and compare.

Notice that when x = -1, f(x) = 0

f(-1) is like substituting x = -1 and get the f(x) value at x = -1.

Thus, f(-1) = 0

Solve the system using a matrix. NEED HELP ASAP!!:):)

Answers

1,-4 sorry if wrong :(

Step-by-step explanation:

1.3x+2y=13

3x+2y+-2y=13+-2y

3x=-2y +13

3x/3=2y+13/3

The answer is x= -2/3 y+13/3

2.x= -2/3 y+ -1/3 z +13/3

3.x= -1/5 y +9/5

You order pizza on Door Dash. It costs $39. Sales tax is 7%. What is the sales tax?

Answers

Answer:

Sales tax is $2.73. Making the total $41.73

Answer:

2.73 or [tex]\frac{273}{100}[/tex]

Step-by-step explanation: