An open top box with a square base has a surface area of 100 square inches. express the volume of the box as a function of the length of the edge of the base. what is its domain?
Answers
The box has a base of area: [tex]a \cdot a = a^2 (square ft) [/tex],
and 4 lateral faces of area ah (square ft).
thus the total area of the box is [tex]a^2+4ah[/tex].
we can reduce the number of variables as follows:
[tex]a^2+4ah=100\\\\4ah=100-a^2\\\\h= \frac{100-a^2}{4a}= \frac{25}{a}- \frac{a}{4} [/tex]
The volume of the box is given by the function:
[tex]V(a)=a \cdot a \cdot (\frac{25}{a}- \frac{a}{4} )=25a- \frac{a^3}{4} [/tex].
To find the domain we solve
[tex]25a- \frac{a^3}{4} \ \textgreater \ 0\\\\a(25- ( \frac{a}{2} )^{2} )\ \textgreater \ 0\\\\a(5- \frac{a}{2})(5+ \frac{a}{2} )\ \textgreater \ 0[/tex]
the roots of the expression are 0, 10 and -10. Since the overall sign of the expression is negative, we have the following sign table:
++++++++++++(-10)----------(0)++++++++++(10)--------------------
thus the solution of the inequation is (-infinity, -10) union (0, 10)
but we have another condition: a>0, since each side must be positive.
combining these 2 conditions, we have the domain: (0, 10)
Answer: [tex]V(a)=25a- \frac{a^3}{4}[/tex], Domain (0, 10)
Given:
An open-top box with a square base has a surface area of 100 square inches.
Question:
Express the volume of the box as a function of the length of the edge of the base. What is its domain?The Process:
Let the length of the edge of the base = x Let height = hPart-1: The surface area
Let us arrange the equation to get the surface area of the box with a square base. Recall that the box is without a lid and its surface area is 100 square inches.
[tex]\boxed{ \ Surface \ area = area \ of \ base + (4 \times area \ of \ rectangle) \ }[/tex]
[tex]\boxed{ \ x^2 + (4 \cdot x \cdot h) = 100 \ }[/tex]
[tex]\boxed{ \ x^2 + 4xh = 100 \ }[/tex]
From the above equation, we set it again so that "xh" is the subject on the left.
Both sides are subtracted by x².
[tex]\boxed{ \ 4xh = 100 - x^2 \ }[/tex]
Both sides are divided by 4.
[tex]\boxed{ \ xh = \frac{100 - x^2}{4} \ }[/tex] ... (Equation-1)
That is the strategy we have prepared.
- - - - - - - - - -
Part-2: The volume
[tex]\boxed{ \ Volume \ of \ the \ box = length \times width \times height \ }[/tex]
[tex]\boxed{ \ Volume = x \cdot x \cdot h \ }[/tex]
[tex]\boxed{ \ Volume = x^2h \ }[/tex] ...(Equation-2)
Substitution Equation-1 into Equation-2.
[tex]\boxed{ \ Volume = x(xh) \ }[/tex]
[tex]\boxed{ \ Volume = x \bigg( \frac{100 - x^2}{4} \bigg) \ }[/tex]
[tex]\boxed{ \ Volume = \frac{100x - x^3}{4} \ }[/tex]
[tex]\boxed{ \ Volume = 25x - \frac{1}{4}x^3 \ }[/tex]
Thus, an expression of the volume of the box as a function of the length of the edge of the base is [tex]\boxed{\boxed{ \ Volume = 25x - \frac{1}{4}x^3 \ }}[/tex]
- - - - - - - - - -
Part-3: The domain of volume
The value of volume must always be positive, i.e., V > 0.
[tex]\boxed{ \ 25x - \frac{1}{4}x^3 > 0 \ }[/tex]
Both sides are multiplied by 4.
[tex]\boxed{ \ 100x - x^3 > 0 \ }[/tex]
Both sides are multiplied by -1, notice the change in the sign of the inequality.
[tex]\boxed{ \ x^3 - 100x < 0 \ }[/tex]
[tex]\boxed{ \ x(x^2 - 100) < 0 \ }[/tex]
[tex]\boxed{ \ x(x - 10)(x + 10) < 0 \ }[/tex]
We get [tex]\boxed{ \ x = 0, \ x = 10, \ and \ x = - 10 \ }[/tex].
Since the values of x cannot be negative, x = -10 are promptly rejected. For x = 0 can be used as one of the domain limits.Consider the test of signs:
x(x - 10) (x + 10) is negative to the left of x = 10, and positive to the right of x = 10 on the number line.
Examples of tests:
[tex]\boxed{ \ for \ x = 2 \rightarrow 2(2 - 10)(2 + 10) < 0 \ }[/tex][tex]\boxed{ \ for \ x = 11 \rightarrow 11(11 - 10)(11 + 10) > 0 \ }[/tex]Remember this form above, [tex]\boxed{\ x(x - 10)(x + 10) < 0 \ }[/tex], the value of the test result must be negative (because < 0).
Thus, the domain of the volume is [tex]\boxed{ \ 0 < x < 10 \ }[/tex] or [tex]\boxed{ \ (0, 10) \ }[/tex]
Learn moreFind out the area of a cube https://brainly.com/question/12613605What is the volume of each prism? https://brainly.com/question/414021The volume of a rectangular prism https://brainly.com/question/11613210Keywords: an open-top box, with, a square base, has a surface area, 100 square inches, express, the volume, as a function, the length, edge, base, what, its domain, test of signs, substitution
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Answers
f₁ = 6
f₂ = f₁ + 5 = 11
f₃ = f₂ + 5 = 16
f₄ = f₃ + 5 = 21
and so on
The sequence is
6, 11, 16, 21, ...,
This is an arithmetic sequence with a common difference of 5.
In function notation, the sequence is
[tex]f_{n+1} = 6 + 5n, \\ \, n=0,1,2, \, ...,[/tex]
This represents the equation of a straight line with n as the independent variable and f as the dependent variable.
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A graph of the line is shown below.
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The sequence is [tex]f_{n+1} = 6 + 5n, \,\, n=0,1,2, \, ...,[/tex]
The slope is 5.
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An automotive repair center charges $45 for any part of the first hour of labor, and $25 for any part of each additional hour. Which of the following is a correct cost?
A. C(t) = 95 for 1 < x ≤ 2
B. C(t) = 95 for 2 < x ≤ 3
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0-1 h= $45
1-2h= $70 ($45 +1*$25)
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3-4h= $120 ($45 +3*$25)
Looking at the list, the answer should be
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The required difference in the financial result is $13000.
given that,
An ice cream company made a profit of $24,000 in 2011. The same company had a loss of $11,000 in 2012. What is the difference between the company's financial results for 2011 and 2012 is to be determined.
In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication, and division,
Here,
In the year 2011, an ice cream company made a profit of $24,000
in the year 2012, the same company had a loss of $11,000.
Difference in the result = 24000 - 11000 = $13,000
Thus, the required difference in the financial result is $13000.
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Answers
x + x + 2 = 170
2x + 2 = 170
2x = 170 - 2
2x = 168
x = 168/2
x = 84
x + 2 = 84 + 2 = 86
so ur 2 numbers are 84 and 86
The required value of integers are 84 and 86 respectively.
How to write a linear equation?A linear equation for the given case can be written by assuming any variable as the unknown quantity. Then, as per the given data the required operations are done and it is equated to some value.
Suppose the first even integer be x.
Then, the second even integer is x + 2.
Since the sum is given as 170, the following equation can be formed as,
x + x + 2 = 170
⇒ 2x + 2 = 170
⇒ 2x = 168
⇒ x = 84
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y = 5x – 1
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one solution: (1, 4)
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Answers
-15x - 3y = 3
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x = 0
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y = 5x - 1...so we sub 5x - 1 in for y in the other equation
-15x - 3y = 3
-15x - 3(5x - 1) = 3
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Solumedrol 1.5 mg/kg is ordered for a child weighing 50 lb. Solumedrol is available as 75 mg / 1 mL is available. How many mL must the nurse administer?
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x = 14/5
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Answers
First, you need to find the quotient of 4 and 13, obviously.
4/13 is equal to roughly 0.308.
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find the solution of
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Answers
Given 2x - 2y = -20 and -2x + 4y = 14, add the second equation to the first:
2x - 2y = -20
-2x + 4y = 14
----------------------------------------------
2y = -6. Solving for y: y = -3.
Find x by subst. y=-3 into either of the given equations.
If we choose the first equation, then it becomes 2x - 2(-3) = -20
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1meter =1x10^9 in sicentific notation
Answers
2 times as much as 4 is
Answers
The answer is 8.
To solve the given mathematical question, we need to calculate twice as much as the number 4. The phrase 2 times as much as indicates multiplication by 2. Therefore, we multiply the number 4 by 2 to find the result.
Mathematically, this can be represented as:
[tex]\[ 2 \times 4 = 8 \][/tex]
So, 2 times as much as 4 is 8.
How many ways are there to pick a starting five from a basketball team of twelve members
Answers
Answer:
95040
Step-by-step explanation:
Given : There are 12 members in basketball team.
To Find: How many ways are there to pick a starting five from a basketball team of twelve members
Solution:
Since we are asked to find no. of ways are there to pick a starting five from a basketball team of twelve members
So, we will use permutation over here.
Formula: [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]
n = 12
r = 5
Substitute the value sin the formula .
[tex]^{12}P_5=\frac{12!}{(12-5)!}[/tex]
[tex]^{12}P_5=\frac{12!}{(7)!}[/tex]
[tex]^{12}P_5=\frac{12 \times 11\times 10 \times 9\times 8 \times 7!}{7!}[/tex]
[tex]^{12}P_5=12 \times 11\times 10 \times 9\times 8[/tex]
[tex]^{12}P_5=95040[/tex]
Hence there are 95040 ways to pick a starting five from a basketball team of twelve members.
There are 95040 different ways to pick a starting five from a basketball team of twelve.
What are permutation and combination?A permutation is an arrangement of things where order matters, AB and BA are two different permutations.
The combination is a selection of things where order does not matter, AB and BA are the same combinations.
Given, The no. of ways to pick a starting five from a basketball team of twelve members.
As stated pick a starting five we'll use permutation.
∴ The no. of ways to pick a starting five from a basketball team of twelve members is [tex]^{12}P_5[/tex].
= 12!/(12 - 5)!.
= 12!/7!.
= 12×11×10×9×8.
= 95040 different ways.
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(hard.) in a certain psychology experiment, each subject is presented with three ordinary playing cards, face down. the subject takes one of these cards. the subject also takes one card at random from a separate, full deck of playing cards. if the two cards are from the same suit, the subject wins a prize. what is the chance of winning
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To answer this question you need to know what the same suit means and how many cards in a deck is. In the standard card, there are 52 cards in a deck. The deck also compromised of 4 suit which was clubs (♣), hearts (♥), spades (♠) and diamonds (♦), Each suit has the same number ratio from those 52 cards, which mean 13 cards for each suit.
In this test, the subject takes one of three cards first. There is no information about this card, so we can assume that it is completely random and having one of the 4 available suits. After that, the subject takes a card from the separate full deck.
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Answers
Hope this helps!
11 1/3 is the answer
When two parallel lines are cut by a transversal, angles A and B are alternate interior angles that each measure 105°. What is the measure of each of the other alternate interior angles?
75°
90°
105°
180°
Answers
Therefore the sum of the angles must be 180.
x + 105 = 180
x = 180 - 105 = 75
The measure of the other alternate angles is 75.
The measure of each of the other alternate interior angles is; 75°
How to find the Alternate Interior Angle?We know that according to Angle theorems, two adjacent alternate interior angles make up a straight line. Sum of angles on a straight line equals 180°.
Thus, if the measure of the other interior angle is x, then we have;
x + 105 = 180
x = 180 - 105
x = 75°
The measure of the other alternate angles is 75°.
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To decrease an amount by 7%, what single multiplier should you use?
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I have 7 hundreds blocks, 5 tens blocks, and 8 ones blocks. I use my blocks to model two 3-digit numbers. What could my two numbers be?
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5 tens blocks = 50
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Trevor earns $7.80 an hour take home pay and works 20 hours each week. Trevor wants to buy a new laptop computer to take to college that costs $780. How many weeks will it take for Trevor to achieve his goal?
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7.89*20= $156 per week
156*5 weeks= $780
Answer: 5 weeks
Step-by-step explanation:
What is 3/8 written as a decimal
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4*2.2046
8.8184 lb.
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4*2.2046
The relationship 1/sin(θ) is equivalent to which of the following trigonometric functions?
A. cot(θ)
B. csc(θ)
C. arcsin(θ)
D. sec(θ)
Answers
Answer:
Option B is the correct answer.
Step-by-step explanation:
We have the following trigonometric relations
[tex]cosec\theta =\frac{1}{sin\theta }\\\\sec\theta =\frac{1}{cos\theta }\\\\cot\theta =\frac{1}{tan\theta }\\\\cot\theta =\frac{cos\theta}{sin\theta }\\\\tan\theta =\frac{sin\theta}{cos\theta }[/tex]
Here cosec θ is the correct answer.
Option B is the correct answer.
Evaluate. Write in simplest form.
6 - b for a = 10, b = –9
--------
3a
A.1/2
B.-/310
C.1/10
D. 2 HELP ME BROS
Answers
6 - (-9)
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3a
3(10)
30
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The recommended weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams.
a. Write and solve an absolute value equation to find the minimum and maximum acceptable soccer ball weights. Use x
as the variable.
The recommended weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams.
a. Write and solve an absolute value equation to find the minimum and maximum acceptable soccer ball weights. Use x
as the variable.
Answers
Actual weight = |Recommended weight +/- 20|
x = |430 +/- 20|
Answer:
The required absolute equation is: |x-430|=20
The minimum and maximum acceptable soccer ball weights is 410 grams and 450 grams respectively.
Step-by-step explanation:
Consider the provided information.
The recommended weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams.
Let x is the actual weight of the soccer ball.
The difference of actual weight and recommended weight can vary by 20 grams.
Thus, the required absolute equation is:
|x-430|=20
x-430=20 or x-430=-20
x=450 or x=410
The minimum and maximum acceptable soccer ball weights is 410 grams and 450 grams respectively.
643=4×160+n in a related division form