A rectangular container with a square base, an open top, and a volume of 4,000 cm³. The minimum surface for the container is 1,200 cm².
The result is obtained by using the derivative.
Determine the minimum surface areaThese are the specified parameters:
Volume container is 4,000 cm³.
Let side of square base = a, and height of the container = h, so:
Volume = a × a × h
4,000 = a² h
h = [tex]\displaystyle \frac{4000}{a^2}[/tex] ... (1)
Surface area with a square base, an open top is area of a square + 4 times the vertical side.
Surface area (A) = a² + 4ah
Substitute the value of h = [tex]\displaystyle \frac{4000}{a^2}[/tex] into surface area.
A = a² + 4ah
A = a² + 4a ([tex]\displaystyle \frac{4000}{a^2}[/tex])
A = a² + [tex]\displaystyle \frac{16000}{a}[/tex]
So that the surface area is minimal, when the derivative and equate to 0.
A = a² + 16000 a⁻¹
A' = 0
2 a²⁻¹ + (-1) 16000 a⁻¹⁻¹
2a - 16000 a⁻² = 0
2a - [tex]\displaystyle \frac{16000}{a^2}[/tex] = 0
2a = [tex]\displaystyle \frac{16000}{a^2}[/tex]
2a³ = 16000
a³ = [tex]\displaystyle \frac{16000}{2}[/tex]
a³ = 8000
a = ∛8000
a = 20
Surface area with a square base, an open top.
Substitute the value of a = 20
A = a² + [tex]\displaystyle \frac{16000}{a}[/tex]
A = 20² + [tex]\displaystyle \frac{16000}{20}[/tex]
A = 400 + 800
A = 1200
The minimum surface for the container is 1,200 cm².
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developing proof: what’s wrong with the triangles? explain.
For the given triangles , developing proof is not possible due to following reasons:
First triangle : 21 + 25 < 48 not true.
Second triangle : Interior angle 130° > Exterior angle 125°
As given in the question,
For the given triangles developing proof is not possible due to following reasons:
First triangle:
Measure of three sides are given as 21, 25 , and 48.
As per the property to form a triangle : Sum of two sides is always greater than the third side.
Here, 21 + 25 < 48 .
Above mention property does not satisfied by given triangle.
Second triangle:
In a triangle, measure of exterior angle is always greater the opposite interior angles.
Here, measure of interior angle 130° > measure of exterior angle 125°
Does not satisfy the relation of exterior angles with interior angle.
Therefore, for the given triangles , developing proof is not possible due to following reasons:
First triangle : 21 + 25 < 48 not true.
Second triangle : Interior angle 130° > Exterior angle 125°.
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Eve is organizing a race to raise money for a local children's hospital. She makes $20 for each 5K participant and $45 for each 10K participant. Her goal is to raise $9,500.
If x represents the number of participants in the 5K race and y represents the number of participants in the 10K race, the linear equation for the given scenario is
x + y = . If Eve raises exactly $9,500, is it possible for there to be 220 participants in the 5K race?
(yes/no)
No It is not possible to have 220 participants in the 5k race.
How to determine the number of participants in the racesInformation from the question
She makes $20 for each 5K participant
$45 for each 10K participant.
x represents the number of participants in the 5K race
y represents the number of participants in the 10K race
Eve raises exactly $9,500, is it possible for there to be 220 participants in the 5K race = ?
From the question
$20 for each 5K participant and x represents the number of participants in the 5K race. = 20x
$45 for each 10K participant and y represents the number of participants in the 10K race = 45y
Eve raises exactly $9,500 means the sum of amount from 5k and 10k is 9500 = 20x + 45y = 9500
if Eve raise 220 participants in the 5k race then the equation will be
20 * 220 + 45y = 9500
4400 + 45y = 9500
45y = 9500 - 4400
45y = 5100
y = 113.333
The participants are human being and cannot be represented as fractions hence 220 is impossible for the 5k race
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If the expression [tex]\sqrt{25x^5y^5}/xy^4[/tex] is written in the form ax^by^c. then what is the product of a, b and c?
The product of a, b, and c is -45/4.
What is multiplication?Multiplication is the process of adding a number up to a given number.
The given expression is [tex]\frac{\sqrt{25x^5y^5}}{xy^4}[/tex].
Rewrite the expression as follows,
[tex]\frac{\sqrt{25x^5y^5}}{xy^4}[/tex]
[tex]=\frac{5x^{\frac{5}{2}}y^{\frac{5}{2}}}{xy^4}\\=5x^{\frac{5}{2}-1}y^{\frac{5}{2}-4}\\=5x^{\frac{3}{2}}y^{-\frac{3}{2}}[/tex]
The values of a, b, and c are 5, 3/2, -3/2.
The product of a, b, and c is,
5 × 3/2 × (-3/2)
= -45/4
Hence, the product of a, b, and c is -45/4.
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HELP NOWWWW LOOK AT IMAGE NOWW
After multiplying the simplified expression is [tex]\frac{-x+5}{2x + 6}[/tex] ; x ≠ -3 , 3 , -2 , -5 , the correct option is (b) .
In the question ,
it is given that ,
the expression is (x²-25)/(2x²+10x+12) × (-x²+x+6)/(x²+2x-15)
we need to find the value of the expression after multiplication .
= (x²-25)/(2x²+10x+12) × (-x²+x+6)/(x²+2x-15)
On splitting the given quadratic equation into factors , we get
= (x+5)(x-5)/2(x+3)(x+2) × -(x+2)(x-3)/(x-3)(x+5)
= [tex]\frac{(x+5)(x-5)}{2(x+3)(x+2)}[/tex] × [tex]\frac{-(x+2)(x-3)}{(x-3)(x+5)}[/tex]
After cancelling out the common factors , we get
= -(x - 5)/2(x + 3)
= -x+5 / 2x + 6
= [tex]\frac{-x+5}{2x + 6}[/tex] : x ≠ -3 , 3 , -2 , -5
Therefore , After multiplying the simplified expression is [tex]\frac{-x+5}{2x + 6}[/tex] ; x ≠ -3 , 3 , -2 , -5 .
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What is the solution of the system? Solve using matrices. ( -3x + 2y = 10 -4x + 3y = 2 (-26 -34) (-34 -26) (26 34) O( no solution)
x + 2y = -1,, x-3y=-2
(-7 ,3)(3,-7) (7,-3)(no solution)
Inverse Matrices and Systems
Answer:
(a) (-26, -34)
Step-by-step explanation:
You want the solution to the system of equations ...
-3x +2y = 10-4x +3y = 2apparently using inverse matrices.
Matrix equationThe system can be represented by the matrix equation ...
AX = B
where A is the 2×2 matrix of coefficients, X is the 2×1 variable vector, and B is the 2×1 matrix of constants.
The solution to this system is found by multiplying on the left by the inverse of A:
A⁻¹·AX = A⁻¹·B
X = A⁻¹·B
The attachment shows the result of making this calculation:
[tex]\boxed{X=\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{c}-26\\-34\end{array}\right]}[/tex]
QUESTION 9/10 HELP ASAPP
Answer:
y = 3x + 8Step-by-step explanation:
GivenThe y-intercept b = 8,Slope m = 3.The equation is y = mx + by = 3x + 8please help and show steps please
Answer:
Step-by-step explanation:
f(x)=3x
2
+9x−16
Quadratic polynomial can be factored using the transformation ax
2
+bx+c=a(x−x
1
)(x−x
2
), where x
1
and x
2
are the solutions of the quadratic equation ax
2
+bx+c=0.
3x
2
+9x−16=0
All equations of the form ax
2
+bx+c=0 can be solved using the quadratic formula:
2a
−b±
b
2
−4ac
. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=
2×3
−9±
9
2
−4×3(−16)
Square 9.
x=
2×3
−9±
81−4×3(−16)
Multiply −4 times 3.
x=
2×3
−9±
81−12(−16)
Multiply −12 times −16.
x=
2×3
−9±
81+192
Add 81 to 192.
x=
2×3
−9±
273
Multiply 2 times 3.
x=
6
−9±
273
Now solve the equation x=
6
−9±
273
when ± is plus. Add −9 to
273
.
x=
6
273
−9
Divide −9+
273
by 6.
x=
6
273
−
2
3
Now solve the equation x=
6
−9±
273
when ± is minus. Subtract
273
from −9.
2 Which ordered pairs represent vertices of the
polygon?
Circle THREE correct answers.
(-11, 13)
(1, -1)
(1/4 , 2 2/1)
(1,-3)
The ordered pairs representing the vertices of a polygon are
(-11, 13) , (1, -1) and (1,-3).
Given, the ordered pairs representing the vertices of the polygon.
as, the given ordered pairs are
(-11, 13) , (1, -1) , (1/4 , 2(2/1)) , (1,-3).
Now, from the given ordered pairs
the pairs which are representing the vertices of a polygon are
(-11, 13) , (1, -1) and (1,-3).
So, the ordered pairs representing the vertices of a polygon are (-11, 13) ,
(1, -1) and (1,-3).
Hence, the ordered pairs representing the vertices of a polygon are
(-11, 13) , (1, -1) and (1,-3).
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Joelle is working on a school report in a word processor. She wants to place an oversized image which is 30.51 cm wide so that it appears centered on the page. The width of the page is 21.59 cm In the program, horizontal positions to the right of the leftmost edge of the page are positive.
What horizontal position for the left edge of the image should Joelle use to center the image?
The horizontal position for the left edge of the image should Joelle use to center the image is -4.46 cm .
In the question ,
it is given that ,
the length of the over sized image = 30.51 cm and
the length of the page = 21.59 cm
to find the margin , we subtract the width of the word processor page from the width of the oversized image.
that is
margin = 30.51 - 21.59 = 8.92 cm
Since , we have margin on two side ,that is horizontal positions to the left most side and horizontal positions to the right of the left most edge of the page . we divide 8.92 by 2 .
= 8.92/2
= 4.46 cm
it is given that the horizontal positions to the right of the leftmost edge of the page are positive,
So , horizontal position for the left edge of the image which Joelle should use to center the image would the negative .
the horizontal position is -4.46 cm .
Therefore , The horizontal position for the left edge of the image should Joelle use to center the image is -4.46 cm .
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A student club plans to buy food for a soup kitchen. A case of vegetables coats $10.68. The club can spend at most $50 for this project. What are all the possible numbers of cases c that the club can buy?
Answer:
0,1,2,3,4
Step-by-step explanation:
[tex]\frac{50}{16.68}[/tex] = 4.68 rounded to the nearest hundredths.
931.9572 Find the digits in the hundreds place, in the hundredths place, and in the ten thousandths place for the following number.
A wooden door costs AED 600 plus a 12% tax. What is the price of the door including the tax?
Answer:
672
Step-by-step explanation:
600+ .12(600)
600 + 72 = 672
7 Résoudre les équations d'inconnue x suivantes:
a.-2x + 7 = -x +9
c.4x +15=2x+6
b. 8x+26=5x+14
d. 7x+31= 16x +17
Answer:
A)
[tex]\sf -2x+7=-x+9[/tex]
[tex]\sf -2x+7-7=-x+9-7[/tex]
[tex]\sf -2x=-x+2[/tex]
[tex]\sf -2x+x=-x+2+x[/tex]
[tex]\sf -x=2[/tex]
[tex]\sf \cfrac{-x}{-1}=\cfrac{2}{-1}[/tex]
[tex]\boxed{\sf x=-2}[/tex]
____________________
B)
[tex]\sf 4x +15=2x+6[/tex]
[tex]\sf 4x+15-15=2x+6-15[/tex]
[tex]\sf 4x=2x-9[/tex]
[tex]\sf 4x-2x=2x-9-2x[/tex]
[tex]\sf 2x=-9[/tex]
[tex]\sf \cfrac{2x}{2}=\cfrac{-9}{2}[/tex]
[tex]\boxed{\sf x=-\cfrac{9}{2}}[/tex]
____________________
C)
[tex]\sf 8x+26=5x+14[/tex]
[tex]\sf 8x+26-26=5x+14-26[/tex]
[tex]\sf 8x=5x-12[/tex]
[tex]\sf 8x-5x=5x-12-5x[/tex]
[tex]\sf 3x=-12[/tex]
[tex]\sf \cfrac{3x}{3}=\cfrac{-12}{3}[/tex]
[tex]\boxed{\sf x=-4}[/tex]
____________________
D)
[tex]\sf 7x+31=16x+17[/tex]
[tex]\sf 7x+31-31=16x+17-31[/tex]
[tex]\sf 7x=16x-14[/tex]
[tex]\sf 7x-16x=16x-14-16x[/tex]
[tex]\sf -9x=-14[/tex]
[tex]\sf \cfrac{-9x}{-9}=\cfrac{-14}{-9}[/tex]
[tex]\boxed{\sf x=\cfrac{14}{9}}[/tex]
What would be the compound interest obtained on an amount of $4800 at a rate of 5% per annum for 3 years?
A $448.7
B) $817.8
C)$623.5
D $756.6
The total compound interest obtained is $756.6.
We are given an initial amount equal to $4,800. This is the principal amount that is denoted by the variable "P". The rate of interest is 5% per annum and is denoted by "r". The interest is compounded annually and denoted by "I". The time duration is 3 years, denoted by the variable "t". Let the final amount be represented by the variable "A". We can use the given formula.
A = P(1 + r)^t
A = $4,800*(1 + 0.05)³
A = $4,800*(1.05)³
A = $4,800*1.157625
A = $5,556.6
The compound interest earned is the difference between the initial and the final amount.
I = A - P
I = $5,556.6 - $4,800
I = $756.6
Hence, the interest amount is $756.6.
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Joe bought 5 1/4-pounds of trail mix for $24.94. What is the price per pound of the trail mix?
Answer:
Step-by-step explanation:
probably close to 5$
Find the angle between the following two vectors.
< 8, 3 > and < 4, −3 >
The angle between the two vectors < 8, 3 > and < 4, −3 > is 57.43°
How to determine the angle between the following two vectors?Given: vectors < 8, 3 > and < 4, −3 >
To find the angle between two vectors, we can use the dot product method:
First, you will compute the dot product and magnitudes of both vectors. Thus,
Let a = < 8, 3 > and b = < 4, −3 >
a · b = <8, 3> ·<4, -3> = 8(4) + (3)(-3) = 32-9 = 23
|a| = √(8² + 3²) = √(64+9) = √73
|b| = √(4² + (-3)²) = √(16 + 9) = √25 = 5
By using the angle between two vectors formula using dot product,
θ = cos⁻¹ [ (a · b) / (|a| |b|) ]
θ = cos⁻¹ (23 / √73 · 5)
θ = cos⁻¹ (23/42.72) = 57.43°
Therefore, the angle between the following two vectors is 57.43°
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Write a linear equation that has m=-4 and passes through (5,9)!
Answer:
[tex]y=-4x+29[/tex]
Step-by-step explanation:
Using point-slope form,
[tex]y-9=-4(x-5) \\ \\ y-9=-4x+20 \\ \\ y=-4x+29[/tex]
A line passes through the points (0, 0) and (–17, –11). What is its equation in slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form
A line passes through the points (0, 0) and (–17, –11). The equation in the slope-intercept form will be y=9/5x+0
What is a straight line?A straight line is a combination of endless points joined on both sides of the point.
The slope 'm' of any straight line is given by:
A linear equation has the form y = mx + b in the slope-intercept format. X and Y are the variables in the equation. The values m and b represent the line's slope (m) and the value of y when x is 0.
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
It is given that, the line passes through the points (0, 0) and (–17, –11).
The slope of the line is,
m = (5+4)/(5-0)
m= 9/5
The equation of the line passes through the point as,
(y-y₁)=m(x-x₁)
(y-0)=9/5(x-0)
5(y-0)=9(x-0)
5y=9x+0
y=9/5x+0
Thus, a line passes through the points (0, 0) and (–17, –11). The equation in the slope-intercept form will be y=9/5x+0
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Please help! ASAP!! i really don't understand
Answer: B
Kyler earns more money in one week. Kyler earns $1,150 and Xiang earns $685 in one week.
Step-by-step explanation:
Have A Good Day!
Which statements about this situation are true?
Answer:
D) Alan was farther away from Madison when he started driving
F) Alan and Andrew started the same distance from Madison
Step-by-step explanation:
Hopefully I got them all!!!
(4,-2) with a scale factor of 1.5
question is on image
Answer:
y= -3/4 x + 8
Step-by-step explanation:
y-2 = -3/4 (x-8)
y-2 = -3/4 x + 6
y = -3/4 x + 6 + 2
y = -3/4 x + 8
karen buys four movie tickets for $\$10.50$ each and two buckets of popcorn for $\$5.50$ each. how many dollars does she spend for the movie tickets and the popcorn? express your answer as a decimal to the nearest hundredth.
Karen spent $42.00 on movie ticket and $11.00 on popcorn.
Firstly calculating the amount spent on movie tickets
Cost of one movie ticket = $10.50
Cost of four movie tickets = 4×10.5
Performing multiplication on Right Hand Side of the equation
Cost of four movie tickets = $42.00
Now calculating the amount spent on popcorn
Cost on one bucket of popcorn = $5.50
Cost of two buckets of popcorn = 2 × 5.50
Performing multiplication on Right Hand Side of the equation
Cost of two buckets of popcorn = $11.00
Hence, the cost spent on movie tickets and popcorn is $42.00 and $11.00 respectively.
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what is the ratio of james wins to james play
Answer:
Step-by-step explanation:
a raft left port a and traveled toward port b. at the same time, a motorboat left port b and traveled toward port a. how many hours after departure will the two meet if the raft travels the distance between ports a and b after 30 hours and the motorboat travels the same distance in 6 horus?
If the raft travels the distance between ports a and b after 30 hours and the motorboat travels the distance in 6 hours. After the departure the two boats will after 5 hours 4 minutes.
Given that,
After leaving the port,
Raft travels the distance between a and b ports = 30 hours
Motorboats travels the distance between a and b ports = 6 hours
The triangle formed by their tracks and the line joining their current positions in a right angled triangle.
According to the Pythagoras Theorem,
a² + b² = c²
⇒ (30)²+ (6)² = c²
⇒ 900 + 36 = c²
⇒ 936 = c²
⇒ c = √936
⇒ c = 30.59
Rounding up 30.59 in 30.6.
Therefore, the boats will the two meet by 30 hours and 6 minutes after they leave the port.
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5. USE STRUCTURE What values of a, b, c,
and d will make the equation have one
solution? Then alter your equation so that
it has no solution. Finally, alter the
equation again so that it has infinitely
many solutions.
ax + b = cx+ 8
Many solutions for a = c and b = 8.
No solution for a = c and b ≠ 8.
One solution for a ≠ c, b can be anything.
What is solution of equations?
Any value of the variable that makes the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation equal is considered a solution to the problem. Finding the solution or solutions to an equation is known as solving it.
The two sides of the equation must be identical for all values of x if
ax + b = cx + 8 has an unlimited number of solutions. This suggests that
a = c and b = 8.
The two sides of the equation cannot be equal for any value of x if
ax + b = cx + 8 has no solutions. The only way this is possible is if
a = c and b ≠ 8, therefore b can be anything other than 8.
If there is exactly one solution to the equation ax + b = cx + 8, then a ≠ c, meaning that 'a' can be anything other than c. In this situation, 'b' is not constrained in any way (b can be anything).
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solve x-6y=9 to find x coordinates
_ _ _ _ _ _ _ _ _ _ _ _ _
| | | _ _ _ | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | _ _ _ _ _ _ | | _ _ _| | | | _ _ _ _ _ _
| _ _ _ _ _ _ _ _| | _ _ _ _ _ _| | _ _ _ _ _ _ _ _|
The slope is 1/6x
You have to re-arranged the problem
x - 6y = 9 --> 6y = x - 9
6y = x - 9
__ ____ ----> y = 1/6x - 3/2
6 6
Which function ha real zero at x = –8 and x = 5?
g(x) = x2 3x – 40
g(x) = x2 3x 40
g(x) = x2 14x – 40
g(x) = x2 14x 40
The function which has real zero at x = –8 and x = 5 is x² + 3x - 40
Given the real zeroes at x = -8 and x = 5.
Factor theorem states that (x-r) is a factor of the polynomial function f(x) if and only if r is a root of the function f(x).
Since, we know that the root of the function i.e g(x) are -8 and 5 then the function has the following factor:
(x+8) = 0 and (x-5) =0
Zero product property states that if ab = 0 if and only if a =0 and b =0.
By zero product property,
(x+8)(x-5) = 0
Now, distribute each terms of the first polynomial to every term of the second polynomial we get;
x(x - 5) + 8(x - 5)
Now, when you multiply two terms together you must multiply the coefficient (numbers) and add the exponent.
x² + 8x - 5x - 40
Combine like terms;
x² + 3x - 40
therefore, the function has real zeros at x = -8 and x =5 is x² + 3x - 40
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Four boxes are sliding with constant speed before each box experiences an unbalanced force of 8 n. Which box would experience the greatest acceleration when the unbalanced force is applied?.
The greatest acceleration when the unbalanced force is applied will be experienced by the least weighted box ( i.e., the mass of 2 kg)
Acceleration: it is defined as rate of change of velocity with respect to time.
Unbalanced Force: it means that the force applied in one direction is greatest than the force applied in the opposite direction.
Given that, F = 8n.
let us assume that the boxes are of 2kg, 4 kg, 6 kg and 8 kg.
According to the 2nd law of motion: the external unbalanced force is directly proportional to rate of change of momentum or Force is equal to the product of mass and acceleration.
F = m × a
⇒ a = F/ m
where, F is Force, m is mass and a is acceleration.
Now, acceleration of the box of 2kg = 8÷2 = 4 m/s²
acceleration of the box of 4kg = 8÷ 4 = 2 m/m²
acceleration of the box of 6 kg = 8÷ 6 = 1.34 m/s²
acceleration of the box of 8kg = 8÷ 8 = 1 m/s²
∴ the box with mass of 2kg experience the greatest acceleration.
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i need help fast please !