From a rectangular piece of cardboard having dimensions a × b, where a = 20 inches and b = 70 inches, an open box is to be made by cutting out an identical square of area x2 from each corner and turning up the sides (see the figure). Express the volume V of the box as a function of x.
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The volume of the box as a function of x will be 1400x - 180x² - 4x³.
First and foremost, you should note that the volume of a rectangular prism is given as:
= Length × Width × Height
where,
Length = 70 - 2x
Width = 20 - 2x
Height = x
Volume = Length × Width × Height
Volume = (x)(20 - 2x)(70 - 2x)
Volume = 1400x - 40x² - 140x² + 4x³
Volume = 1400x - 180x² - 4x³
Therefore, the volume of the box as a function of x will be 1400x - 180x² - 4x³.
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Related Questions
720,808 in expanded form with exponents
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You'll need to determine how many times 2 is a factor of 720808, how many times 3 is a factor, and so on.
Suppose that 2 were a factor 6 times (just an example).
You'd write this particular factor of 720808 as 2^6.
The number 720,808 in expanded form with exponents is written as:
[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]
To express 720,808 in expanded form with exponents, We break it down by place value and then represent each term using powers of 10. The breakdown is as follows:700,000 : [tex]7 \times 10^5 \\[/tex]
20,000 : [tex]2 \times 10^4 \\[/tex]
800 : [tex]8 \times 10^2 \\[/tex]
8 : [tex]8 \times 10^0[/tex]
When we combine these, we get the expanded form:[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]
This representation helps in understanding the value of each digit based on its position in the number.
is a integer a rational number
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How many gallons of a 80% antifreeze solution must be mixed with 60 gallons of 30% antifreeze to get a mixture that is 70% antifreeze?
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(x)(80%) + (60)(30%) = (x+60)((70%)
0.8x + 18 = 0.7x +42
0.8x - 0.7x = 42-18
0.1x = 24
and x = 240 Gal.
Estimate sin pi/3 if sin pi/2 = 1 and sin 0 = 0 and sin pi = 0
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sin(π/3) = √3 / 2
To estimate the value of sin pi/3, which is roughly 1.047 or 60 degrees, we can use standard trigonometric values. Since pi/3 is closer to pi/2 than pi, where sin achieves its maximum of 1, the value of sin pi/3 should be closer to 1 than 0. Standard trigonometric values confirm that sin pi/3 = sqrt(3)/2 = 0.866.
Explanation:To estimate sin pi/3, we can make use of some standard values in trigonometry. Pi is roughly 3.14, hence pi/3 is roughly 1.047. The sin function gives a result of 1 at pi/2 (or 90 degrees), 0 at 0, and again 0 at pi (or 180 degrees). Thus, it achieves its maximum at pi/2 and then decreases to 0 at pi. Intuitively, since pi/3 (or 60 degrees) is closer to pi/2 than to pi, the sin value at pi/3 should be closer to 1 than to 0. From the standard trigonometric values, we know that sin pi/3 = sqrt(3)/2 which is roughly 0.866.
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what are two practical products you could add to find 990 x 37 ; 297 693 2970 6930 69300 6330 29700
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990 x 37 = 36630
Test the given numbers to find the pair that adds up to 36630.
Test 297.
The missing pair will be
36630 - 297 = 36333
Will not work
Test 693.
36630 - 693 = 35937
Will not work
Test 2970.
36630 - 2970 = 33660
Will not work
Test 6930
36630 - 6930 = 29700
This works, because 6930 + 29700 = 36630.
Answer: 6930 and 29700.
One book cost £1.99 how much do four books cost
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[tex]4(2-0.01) = 8 - 0.04 = 7.96[/tex]
A restaurant uses 44 pounds of potatoes for a party of 1212 people. If they are cooking potatoes for a party of 60 people, what should they do?
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Find y'' by implicit differentiation. 5x3 + 2y3 = 1
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The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).
What is implicit differentiation?
In implicit differentiation, we differentiate each side of an equation with two variables by treating one of the variable as a function of other.
Given that;
The equation is;
5x³ + 2y³ = 1
Now, Apply implicit differentiation rule we get;
5x³ + 2y³ = 1
After differentiation;
15x² + 6y² dy/dx = 0
dy/dx = - 15x² / 6y²
dy/dx = -5x²/2y²
Again differentiation;
d²y/dx² = -5/2 (y² × 2x - x² × 2y dy/dx) / y⁴
d²y/dx² = -5/2 (2xy² - 2x²y dy/dx) / y⁴
Substitute value of dy/dx;
d²y/dx² = - 5/2 (2xy² - 2x²y × -5x²/2y²) / y⁴
d²y/dx² = - 5/2 (2xy² + 5x⁴/y) / y⁴
d²y/dx² = - 5x/y² - 25x⁴/2y⁵
y'' = - 5x/y² - 25x⁴/2y⁵
Thus, The value of y'' will be (- 5x/y² - 25x⁴/2y⁵).
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Generate a process that would generate random numbers between –1 and 1 for x and y positions of 10000 points and count how many points fall with the circle, then determine an estimated version of π. (see below to generate floating point random numbers. this generation is not needed to describe the process, but to implement the process)
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import random
import math
count_inside = 0
for count in range(0, 10000): d = math.hypot(random.random(), random.random())
if d < 1: count_inside += 1
count += 1
print 4.0 * count_inside / count
The question is about using the Monte Carlo method to estimate the value of π, by generating random points in a square, counting how many fall inside a unit circle, and finally using this count to calculate π.
Explanation:This question involves the implementation of a famous problem in computational mathematics known as the Monte Carlo method for estimating the value of π:
First, initialize a variable, say 'count', to 0. This variable will keep track of the number of points falling within the circle.Next, carry out a 'for' loop to generate 10000 points. Each point will have x and y coordinates, each a random floating point number between -1 and 1.For each generated point (x, y), calculate the distance from the point to the origin (0,0), which is sqrt(x^2 + y^2). If this distance is less than or equal to 1, it means that the point lies within the unit circle, so increment the 'count' variable by one.After all 10000 points have been generated and tested, 'count' will hold the total number of points lying within the circle. To determine an estimation of π, use the formula 4 * (count / total_number_of_points).This process is based on the mathematical fact that the ratio of the area of a unit circle to the square it is inscribed in is π/4.
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translate into an algebraic expression: the difference of six times a number and 8
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The equation 22 + 0.15p =c
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22 + 0.15p =c could be solved for p; that's one thing you could do now.
Subtracting 22 from both sides, 0.15p = c - 22
Dividing both sides by 0.15, p = (c-22)/0.15 , or p = (20/3)(c-22).
SIMPLIFY the expression! Please!
Answers
1+3^2 = 1+9 = 10
10/5 = 2
2-6+2 = -4+2
answer is -2
Find the equation of the curve that passes through the point 0, 1 6 and whose slope at each point (x, y) is − x 12y .
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The equation is [tex]\(x^2 + 12y^2 - 3072 = 0\),[/tex] obtained from integrating [tex]\(12y \, dy = -x \, dx\)[/tex] with the initial condition.
To find the equation of the curve passing through the point (0, 16) with a slope given by [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \),[/tex] you can use separation of variables.
Given that [tex]\( \frac{{dy}}{{dx}} = -\frac{x}{12y} \)[/tex], we can rewrite it as:
[tex]\[ 12y \, dy = -x \, dx \][/tex]
Now, integrate both sides:
[tex]\[ \int 12y \, dy = \int -x \, dx \]\[ 6y^2 = -\frac{1}{2}x^2 + C \][/tex]
Given that the curve passes through the point (0, 16), substitute these values into the equation:
[tex]\[ 6(16)^2 = -\frac{1}{2}(0)^2 + C \]\[ 6(256) = C \]\[ C = 1536 \][/tex]
Therefore, the equation of the curve is:
[tex]\[ 6y^2 = -\frac{1}{2}x^2 + 1536 \][/tex]
This can also be written in standard form as:
[tex]\[ x^2 + 12y^2 - 3072 = 0 \][/tex]
So, the equation of the curve is [tex]\( x^2 + 12y^2 - 3072 = 0 \).[/tex]
Can the sum of two negative numbers ever be positive?
Answers
two negatives equal positive, and one negative and one positive is i think negative
What is the standard form of 14 ten thousands + 12 thousands
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hope this helps
A car travels 182 miles in 3 hours and 15 minutes. How many miles does it travel per hour?
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[tex]15min*\frac{1hour}{60min}=0.25hr[/tex]
Then we just use the formula:
[tex]d=182*0.25[/tex]
182 / 3.25 = 56 miles per hr
Find the degree of the term -5x squared
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The degree of the term is 2, or quadratic
hope this helps
Let a be a rational number and b be an irrational number. Which of the following are true statements?
A. The sum of a and b is never rational
B. The product of a and b is rational
C. b^2 is sometimes rational
D. a^2 is always rational
E. √a is never rational
F. √b is never rational
Answers
Is a measure of dispersion that indicates how much scores in a sample vary around the mean of the sample?
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Standard deviation is a mathematical measure that is used to evaluate the difference between the mean value of a set of number and each value in the group. It is calculated as the square root of variance by determining the variation between each data point relative to the mean.
caryssa makes bracelets to sell at craft fairs. The supplies she needed to start her business cost $45. In addition, the supplies for each bracelet cost her $3. She sells her bracelets for $5. What is the minimum number of braclets she must sell to make a profit?
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Shantel is studying for her history test. After 20 minutes she is 1/4 of the way done. How much longer will she study?
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Shantel will need to study for 60 minutes more to complete her history review.
Shantel is studying history and after 20 minutes, she is [tex]\frac{1}{4}[/tex] of the way done.
To determine how much longer she will study, we need to calculate how much time is needed to complete the remaining [tex]\frac{3}{4}[/tex] of her study time.
[tex]\frac{1}{4}[/tex] of the study done in 20 minutes means she will study for 4 * 20 = 80 minutes in total.
To find out how much longer she needs to study, subtract the time she has already studied (20 minutes) from the total study time (80 minutes): 80 - 20 = 60 minutes.
Therefore, Shantel will need to study for 60 minutes more to complete her history review.
Between 2004 and 2010, the growth rate of the Florida Panther population was approximately 4%. What was the average annual growth rate?
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Suppose John opens a savings account with $1,000 that compounds interest daily. The APR at the time John deposits the account is 3.5%. He makes no withdrawals or deposits. What is his APY to the nearest hundredth of a percent after 1 year?
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Find the quotient of 5/68
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Why is it impossible to list the set of rational numbers using the roster method?
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There is no way to list all the members of the set of rational numbers.
Solve for W
A=2Lw+2Lh+2wh
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A - 2LH = 2LW + 2WH...factor out w on the left side
A - 2LH = W(2L + 2H) ....divide both sides by (2L + 2H)
(A - 2LH) / (2L + 2H) = W
Lean wants to giftwrap a present she got for her little brother. how many square inches of giftwrap will be needed to cover a box that is 5in × 7in × 3in?
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so.. notice, the box for the gift is really just a rectangular prism, which is really just 6 rectangles stacked up to each other at the edges.
the front and back are two 5x3 rectangles
the left and right are just two 3x7 rectangles
and the top and bottom are just two 5x7 rectangles
so, just get the area of each, sum them up, and that's the area of the rectangular prism and thus of the box, and that's how many in² she will need.
how many gallons will it take to mow a lawn if you maintain the rate of 1/2 gallon of gasoline per 1/4 mowed
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[tex]\bf \begin{array}{ccll} gallons&mowed\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ \frac{1}{2}&\frac{1}{4}\\\\ g&\frac{4}{4} \end{array}\implies \cfrac{\frac{1}{2}}{g}=\cfrac{\frac{1}{4}}{\frac{4}{4}}\implies \cfrac{\frac{1}{2}}{\frac{g}{1}}=\cfrac{\frac{1}{4}}{\frac{4}{4}}\implies \cfrac{1}{2}\cdot \cfrac{1}{g}=\cfrac{1}{4}\cdot \cfrac{4}{4} \\\\\\ \cfrac{1}{2g}=\cfrac{1}{4}\implies 4=2g\implies \cfrac{4}{2}=g\implies 2=g[/tex]
write each rate as a fraction in lowest terms.
4.20 for 36 potatoes
Answers
If so, write the fraction
$4.20
----------------
36 potatoes
and reduce it. 6 divides into both $4.20 and 36 without a remainder:
$0.70 and 6
Thus, the "rate" is 0.70 dollar for 6 potatoes, or 0.35 dollar for three.
If you wish you could rewrite this partially in words:
"Potatoes sell for $0.35/(3 potatoes)," or $0.1166 per potato, or about 12 cents per potato.
How many elements are in the set (A,B,C)
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A has 25 elements
B has 30 elements
C has 35 elements