The surface area of the largest sphere that could fit into the rectangular box of dimensions 12 in x 14 in x 18 in is 324π inches²
The surface area of sphere = 324π inches²
What is a Sphere?
A sphere is symmetrical, round in shape. It is a three dimensional solid, that has all its surface points at equal distances from the center. It has surface area and volume based on its radius. It does not have any faces, corners or edges.
The Surface Area of a Sphere = 4πr²
The Volume of a Sphere = ( 4/3 ) πr³
where r is the radius of the sphere
Given data ,
Let the surface area of the Sphere be = S
Let the dimensions of the rectangular box be L x W x H
12 inches x 14 inches x 18 inches
Now , for the surface area of the sphere to have the largest surface area , the radius should be maximum
From the rectangular box , we can see that the height of the rectangle is best suited as the diameter of the sphere
So , Height of the Box = Diameter of the Sphere
And , Radius of the sphere = Height of the Box / 2
= 18 / 2
Radius of the sphere = 9 inches
Now ,
Surface Area of a Sphere = 4πr²
r = 9 inches
Substituting the values in the equation , we get
Surface Area of a Sphere S = 4 x π x ( 9 )²
= 4 x π x 81
= 324π inches²
Therefore , the value of S is 324π inches²
Hence , The surface area of sphere = 324π inches²
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For how many values of x is h(x) = 0? Justify you answer.
Answer:
the answer for x when h(x) is 2.9
the length of a rectangle is 2 feet more than the width. Find the minimum dimensions if the perimeter is more than 16 feet and the length and width are integers.
Answer:
L = 6 feet ; W = 4 feet
Step-by-step explanation:
l = 2 + w
perimeter/2 = 8
l + w > 8
2 + w + w > 8
2 + 2w > 8
2w > 8-2
2w > 6
2/2 w > 6/2
w > 3
smallest integer > 3 = 4
W= 4
L = 4 + 2 = 6
Simplify 5^3 x 5 please
Answer:
5^4
Step-by-step explanation:
5^3x5
5x5x5x5
5^4
Hopes this helps please mark brainliest
The simplified form of the expression, 5³.5 will be 625.
What is an integer exponent?In mathematics, integer exponents are exponents that should be integers. It may be a positive or negative number. In this situation, the positive integer exponents determine the number of times the base number should be multiplied by itself.
It is given that the expression is, 5³.5
=5³.5
The base should remain constant and the exponents added when multiplying like bases. Keep the base constant when raising a base with power to another power, and multiply the exponents instead. Keep the base constant and deduct the exponents in the denominator and numerator when dividing like bases.
Apply the law of exponent as
xᵃ × xᵇ = xᵃ⁺ᵇ
=5³⁺¹
=5⁴
=5x5x5x5
=625
Thus, the simplified form of the expression, 5³.5 will be 625.
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I NEED HEEEEELP!!
There are 5 cockroaches crawling on each square meter of the front of this building. How many cockroaches are crawling on the front of the building?
Please, show your work and explain. TYSM!! ^^
There are 855 cockroaches crawling on the front of the building
What is an equation?An equation is an expression showing the relationship between two or more numbers and variables.
The population density is given as:
Population density = total animals / area occupied
Given that, the population density of the cockroaches is 5 cockroaches per square meter.
The area of the building = area of triangle + area of square
Area of building = (1/2) * (24 - 9)(12) + (9 * 9) = 90 + 81 = 171 m²
Population density = total animals / area occupied
5 cockroaches per square meter = Total cockroaches / 171 m²
Total cockroaches = 855
There are 855 cockroaches
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Helpi really need this
Answer:
The height of the flagpole is about 25.6 feet.
Step-by-step explanation:
The figure is omitted--please sketch it.
Put your calculator in degree mode.
Let h be the height of the flagpole.
[tex] \tan(52) = \frac{h}{20} [/tex]
[tex]h = 20 \tan(52) = 25.6[/tex]
What is the largest multiple of 30 which is less than 520?
The largest multiple of 30 that is less than the number, 520, can be found to be 510.
How to find the multiple?To find the largest multiple that 30 has, which is also less than 520, first find the number of times that 30 can go into 520:
= 520 / 30
= 17.333
Then use this number to find the largest multiple below 520 by rounding 17.33 downwards to 17.
The multiple of 30 that is the largest and below 520 is then:
= 30 x Rounded down result of division of 520
= 30 x 17
= 510
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A bus ticket used to cost $2.50
Today, 1 bus ticket costs $2.75
The increase in cost is?
The percent increase is?
Determine whether the percent of change is a percent of increase or a
percent of decrease. Then find the percent of change.
%
original: 20
new: 18
by
Answer:
the change is in %age of decrease of 10%
Step-by-step explanation:
original value = 20
new value = 18
decrease = 2
%age decrease = decrease/original value * 100
= 2/20 * 100
= 10%
Emmett is planning a party at an escape room. The escape room will charge Emmett a
booking fee and an additional cost per person attending the party.
This situation can be modeled as a linear relationship.
Answer:
B. After the booking fee the party will cost $20 per person
Determine the length of the line segment shown. graph of line segment from negative 6 comma negative 5 to 0 comma 3 100 units 25 units 10 units 8 units
The length of the line segment will be 10 units.
What is Distance?
The distance between two points (x₁ , y₁) and (x₂, y₂) is defined as;
D = √( y₂ - y₁)² + (x₂ - x₁)²
Given that;
Two endpoints of the line segment are (6, -5) and (0, 3).
Now,
The length of the line segment is distance between two endpoints (6, -5) and (0, 3).
So, The distance between two endpoints (6, -5) and (0, 3) is calculated as;
D = √(0 - 6)² + (3 - (-5))²
D = √36 + 64
D = √100
D = 10 units
Thus, The length of the line segment will be 10 units.
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A business started the year with a net worth of $4,873. During the first six months of the year, the business
recovered and increased its net worth to a value of $15,866. Find the average monthly increase in net worth for the 6-month period.
The average monthly increase in net worth for the 6-month period is $1832.16.
Given:
A business started the year with a net worth of $4,873. During the first six months of the year, the business recovered and increased its net worth to a value of $15,866.
Six months net worth = 15,866 - 4873
= 10,993
Average monthly increase in net worth = six month net worth / 6
= 10,993/6
= $1832.16.
Therefore The average monthly increase in net worth for the 6-month period is $1832.16.
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Which of the following equations are equivalent? Select three options. 2 + x = 5 x + 1 = 4 9 + x = 6 x + (negative 4) = 7 Negative 5 + x = negative 2
A fair six-sided die is rolled twice. What is the theoretical probability that the first number that comes up is greater than or equal to the second number?
OA. 1
6
OB. 1
2
O c. 7
12
O
D. 5
Answer:
c. 7/12
Step-by-step explanation:
There are 36 possible outcomes when you roll a die twice.
11, 12, 13, 14, 15, 16
21, 22, 23, 24, 25, 26
31, 32, 33, 34, 35, 36
41, 42, 43, 44, 45, 46
51, 52, 53, 54, 55, 56
61, 62, 63, 64, 65, 66
The desired outcomes are the ones whose first number is greater than or equal to the second number. They are shown in bold and underline above.
There are 21 desired outcomes.
p(first number ≥ second number) = 21/36 = 7/12
Answer: c. 7/12
Distance in Feet
1. Earlier this week the famous "Live Cannonball” - Gunter Money was shot from a cannon at the Austin Expo.
Gunter flew to a height of 70 feet and traveled a distance of 160 feet only to land perfectly in a safety net.
Assume (0, 0) was the starting point for Gunter's flight and that he landed in the net at (160, 0).
(a) Use matrices to find the equation for Gunter Money's path.
(b) Write the equation for Gunter Money's path in vertex form.
(c) device a method to estimate the angle at which Gunter money takes off
The 160 feet range and 70 feet height of the path of Gunter Money which can be described with a quadratic equation, gives;
(a) The path is; y = -1.09375×10²·x² + 1.75·x
(b) The vertex form of the equation is; y = -1.09375·(x - 80)² + 70
(c) The angle at the start is approximately 60.3°
What is quadratic equation?A quadratic equation is a single variable polynomial equation of the second degree.
The points in the path which is a path of a projectile and has a shape of a parabola are;
(0, 0), (80, 70), and (160, 0)
The equations are;
y₁ = a·x₁² + b·x₁ + c
y₂ = a·x₂² + b·x₂ + c
y₃ = a·x₃² + b·x₃ + c
Plugging in the coordinates of the three points in the path gives;
0 = a·0² + b·0 + 1·c70 = a·80² + b·80 + 1·c0 = a·160² + b·160 + 1·cThe matrix becomes;
[tex]A = \begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}[/tex]
[tex]X = \begin{bmatrix}a \\b \\c\end{bmatrix}[/tex]
[tex]Y = \begin{bmatrix}0 \\70 \\0\end{bmatrix}[/tex]
We have;
A·X = Y, which gives;
[tex]\begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}\cdot \begin{bmatrix}a \\b \\c\end{bmatrix} = \begin{bmatrix}0 \\70 \\0\end{bmatrix}[/tex]
[tex]X = \dfrac{Y}{A} = A^{-1}\cdot Y[/tex]
Where;
A⁻¹ = The inverse of the matrix A
[tex]\begin{bmatrix}a \\b \\c\end{bmatrix} = \frac{ \begin{bmatrix}0 \\70 \\0\end{bmatrix}}{\begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}} = \begin{bmatrix}\dfrac{1}{12800} & -\dfrac{1}{6400} & \dfrac{1}{12800} \\\\-\dfrac{3}{160} &\dfrac{1}{40} & -\dfrac{1}{160} \\\\1 &0 &0 \\\end{bmatrix}\bullet \begin{bmatrix}0\\ \\70\\ \\0\end{bmatrix}=\begin{bmatrix}-\dfrac{7}{640} \\ \\\dfrac{7}{4} \\ \\0\end{bmatrix}[/tex]
[tex]\begin{bmatrix}a \\b \\c\end{bmatrix} = \begin{bmatrix}-\dfrac{7}{640} \\ \\\dfrac{7}{4} \\ \\0\end{bmatrix}[/tex]
[tex]a = -\dfrac{7}{640}[/tex]
[tex]b = \dfrac{7}{4}[/tex]
c = 0
The equation for Gunter Money's path is therefore;
[tex]y = -\dfrac{7}{640} \cdot x^2 + 1\dfrac{3}{4} \cdot x + 0 = -1.09375\times 10^2 \cdot x^2 + 1.75\cdot x[/tex]
(b) The vertex of the path is (h, k) = (80, 70)
The vertex form of a quadratic equation is; y = a·(x - h)² + k
The equation of Gunter Money's path in vertex form is therefore;
[tex]y = -\dfrac{7}{640} \cdot \left(x - 80)^2 + 70 = -1.09375\times 10^2\cdot (x - 80)^2+70[/tex]
(c) The angle of the parabolic path of Gunter Money's motion at a point is given by the value of the arctangent of the differentiation of the function for the motion at the point as follows;
[tex]y' = -\dfrac{7\times 2}{640} \cdot x + 1\dfrac{3}{4}[/tex]
At the point Gunter Money takes off, (0, 0), we have;
[tex]tan(\theta) = y' = -\dfrac{7\times 2}{640} \times 0 + 1\dfrac{3}{4} = 1\dfrac{3}{4} = 1.75[/tex]
[tex]\theta = arctan(1.75) \approx 60.3^{\circ}[/tex]
The angle is approximately 60.3°
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Use the order of operations to simplify this expression (1/4 + 7/8)x 1/2
Order of operation to simplify this expression as follows:
[tex]((1\div4+7\div 8)\times 1)\div 2[/tex]
⇒ [tex]((0.25+7\div 8)\times 1)\div 2[/tex]
⇒ [tex]((0.25+0.875)\times 1)\div2[/tex]
⇒ [tex](1.125\times 1)\div2[/tex]
⇒ [tex]1.125\div 2[/tex]
⇒ [tex]0.5625[/tex]
What is order of operation?
The best way to solve an expression is from left to right if all the operations are the same (for instance, only addition, only subtraction, only multiplication, or only division). However, we must adhere to the order of operations for expressions including several operations.
The rule that dictates how to solve an expression involving numerous operations is known as the order of operations.
PEMDAS can help you remember that sequence. In PEMDAS, each letter denotes a mathematical operation.
Given Expression is ; [tex]((1\div4+7\div 8)\times 1)\div 2[/tex]
Firstly we calculate the highest priority between(+,-, x, and [tex]\div[/tex])
Here, division is the highest priority ,
[tex]((0.25+7\div 8)\times 1)\div 2[/tex]
[tex]((0.25+0.875)\times 1)\div2[/tex]
and then we solve parathesis,
[tex](1.125\times 1)\div2[/tex]
now we do multiply,
[tex]1.125\div 2[/tex]
and the final answer will be [tex]0.5625[/tex]
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$6000 are invested in a bank account at an interest rate of 6 percent per year.
Find the amount in the bank after 11 years if interest is compounded annually.
Find the amount in the bank after 11 years if interest is compounded quarterly.
Find the amount in the bank after 11 years if interest is compounded monthly.
Finally, find the amount in the bank after 11 years if interest is compounded continuously.
The value after 11 years with annual compounding is $11,389.79
The value after 11 years with quarterly compounding is $11,552
The value after 11 years with monthly compounding is $11,589.68
The value after 11 years with continous compounding is $70,080.78.
What is the future value?When an amount is compounded annually, it increases in value once a year. When an amount is compounded quarterly, it increases in value four times a year. When an amount is compounded monthly, it increases in value twelve times a year. When an amount is compounded continuously, it increases continuously over the investment period.
The formula for calculating future value:
FV = P (1 + r)^nm
Where:
FV = Future value P = Present value R = interest rate - annual rate / number of compounding m = number of compoundingN = number of yearsValue after 11 years with annual compounding: $6,000 x (1.06)^11 = $11,389.79
Value after 11 years with quarterly compounding: $6,000 x (1 + 0.06 /4)^(11 x 4) = $11,552
Value after 11 years with monthly compounding: $6,000 x (1 + 0.06 /12)^(11 x 12) = $11,589.68
The formula for calculating future value when there is continuous compounding is : A x e^r x N
Where:
A= amount e = 2.7182818 N = number of years r = interest rateValue after 11 years: 6,000 x (2.718^0.06) x 11 = $70,080.78
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What do you observe? What do you wonder? What are 2 things you know and 1 thing that you're missing?
The observation of the graphing speed versus time is given below.
What is acceleration?The rate at which the speed is changing is known as acceleration.
From the graph, the line segment OA and DE have linear acceleration which increases the velocity.
Similarly, from the graph, the line segment BC and EF have linear deacceleration which decreases the velocity.
From the graph, line segment AB has constant acceleration which increases the velocity.
And from the graph, the line segment CD has zero acceleration which makes the velocity to be constant.
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Select the two values of x that are roots of this equation. 2x^2 + 7x + 6 = 0  A. x = –2  B.   C. x = –4  D. x = –3
The root of the quadratic equation is 2x² + 7x + 6 is x = -2 and x = -3 / 2.
How to find the roots of an equation?The roots of a quadratic equation are the values of the variable that satisfy the equation. They are also known as the "solutions" or "zeros" of the quadratic equation.
The equation to find the roots, is a quadratic equation.
Hence,
2x² + 7x + 6
2x² + 3x + 4x + 6
x(2x + 3) + 2(2x + 3)
Hence,
(x + 2)(2x + 3) = 0
Therefore,
x + 2 = 0 or 2x + 3 = 0
x = -2 or x = -3 / 2
The roots of the equation is x = -2 and x = -3 / 2
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Fine the slope of the line and use rise/run to fill in the blanks
Answer: Rise is 5 and run is 1 and now that is equal 5
Step-by-step explanation:
The diameter of a semicircle is 4 yards. What is the semicircle's perimeter?
Use 3.14 for x.
d=4 yd
The diameter of a semicircle is 4 yards. The perimeter for a semicircle having a diameter of 4 yards is 10.28 yards.
The perimeter of a shape is the distance of the path or boundary that surrounds the shape.
The formula for the perimeter of a semicircle is πr + 2r
π(pie)=3.14
r(radius)=2 yards (∴ radius=1/2diameter)
∴ the perimeter of a semicircle = πr + 2r
= 3.14×2+2×2
=10.28 yards.
∴ The perimeter of a semicircle with a diameter of 4 yards is 10.28 yards.
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48 - (3 c+4)=4(c+5)+3 c=???
Answer in multiple forms depending on what needed
Answer:
c = 3
Step-by-step explanation:
48- (3c+4)=4(c+5)+3
48-3c-4=4(c+5)+3 Distributed the - sign
48 -3c -4= 4c+20 +3 Distributed the 4
44-3c =4c+23 Combined like terms
21-3c=4c Subtract 23 from both sides
21=7c Add 3c to both sides
3=c
Write an equation of the line that is PARALLEL to 12x - 4y = 4 and through the point (-9,-2).
Answer:
y=3x+25
Step-by-step explanation:
we need start by making slope intercept form
lets move the 12x after moving it we get
-4y=-12x+4 we need to remove the - 4 from the y so we need to divide the whole equation by - 4
we get y=3x-1
now we can create an equation to get the parallel line
y+2=3(x+9) now distribute
y+2=3x+27 and subtract the two
y=3x+25
Consider the equation and the following ordered pairs: (6,y) and (x,1).
y=−2x+5
The values of y and x are -7 and 2 respectively, completing the ordered pairs: (6,-7) and (2,1).
What are the values of y and x?Given the equation in the question;
y = −2x + 5Ordered pairs: (6,y) and (x,1)To find the y-value that completes the ordered pair ( 6,y ), replace all occurrence of x with 6 in the equation and solve for y.
y = −2x + 5
y = −2( 6 ) + 5
Multiply -2 and 6
y = -12 + 5
y = -7
Hence, the output value is -7, this forms an ordered pair of (6,-7).
To find the x-value that completes the ordered pair ( x,1 ), replace all occurrence of y with 1 in the equation and solve for x.
y = −2x + 5
1 = −2x + 5
Add 2x to both sides
1 + 2x = −2x + 2x + 5
1 + 2x = 5
2x = 5 - 1
2x = 4
x = 4/2
x = 2
Hence, the input value is 2, this forms an ordered pair of (2,1).
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Select the correct answer from each drop-down menu. Using the image below, select the best answer from each pull-down to complete the statements. The measure of ∠FED is ? °. ∠CBD and ∠ ? are complementary angles. Please help. thank you
The value of ∠FED is 120° and ∠CBD is complementary to ∠FDE
What are complementary angles?Complementary angles are two angles that sum up to 90°
For example 60° and 30° are complementary.
Either angle is called the complement of the other angle.
∠FED, ∠FDE and ∠DFE are in a triangle.
so,
∠FED + ∠FDE + ∠DFE = 180°
16x - 8 + 2x + 9 + 35 = 180 ( sum of angles in a triangle)
18x +36 = 180
18x = 180 - 36
18x = 144
x = 144/18
x = 8
so ∠FED = 16x - 8
Substitute x = 8 into the equation above we have
16(8) - 8 = 128 - 8 = 120°
∠FDE = 2x + 9
∠FDE = 2(8) + 9 = 25°
∠BED = 6x + 12
∠BED = 6(8) + 12
∠BED = 48 + 12
∠BED = 60°
∠CBD = -2x + 81
∠CBD = -2(8) + 81
∠CBD = 65°
so since ∠CBD = 65° and 90 - 65 = 25°
∠CBD is complementary to ∠FDE
In conclusion, ∠FED is 120° and ∠CBD is complementary to ∠FDE
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Dave’s Furniture Store is having a 35% off sale on tables. Robert paid $195 for a table that was on sale. What was the original price of the table?
Answer:
557.143
Step-by-step explanation:
195 / 35% = 557.14
Help Need ASAP
Paula makes purses. The materials cost $2.25 per purse. She sells them for $5.00 each. Paula wants to invest $5000 for some new equipment. Which of these could be used to determine x, the number of purses Paula must sell to pay for the new equipment?
Responses
5.00x - 2.25x = 5000
5.00x + 2.25x = 5000
5.00x = 5000
2.25x = 5000
The equation that represents the number of purses Paula must sell to pay for the new equipment is 5.00x = 5000.
What is a function?A function can be defined as the outputs for a given set of inputs.
The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.
Given, Paula makes purses and the materials cost $2.25 per purse.
She sells them for $5.00 each now she wants to invest $5000 in some new equipment and we are assuming the no. of purses to be x.
Here the equation 5.00x - 2.25x = 5000 represents Paula's profit.
The equation that represents the no. of purses she must sell to pay for the new equipment is 5.00x = 5000.
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Answer: 5.00x - 2.25x = 5000
Step-by-step explanation:
y5 • y14 =
A. 2y19
B. y60
C. y19
D. 2y60
ALGEBRA
How to solve: 24 - (5 + 3)^2 + 4 × 2^3 step by step.
The value of the exponent given as 24 - (5 + 3)^2 + 4 × 2^3 is -8.
What is an exponent?An exponent is the number of times that a number is multiplied by itself. It should be noted that the power is an expression which shows the multiplication for the same number. For example, in 6⁴ , 4 is the exponent and 6⁴ is called 6 raise to the power of 4.
In this case, the value given will be expressed thus:
= 24 - (5 + 3)^2 + 4 × 2^3
= 24 - (8)² + 4 × 8
= 24 - 64 + 32
= -8
The value is -8.
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A harbor seal dives 35 meters below the ocean's surface. Then it dives down an additional 12 meters Determine the seal's new position relative to the surface of the ocean
The seal's new position relative to the surface of the ocean is 47 meters below.
How to calculate the value?It is important to note that that question has to do with elevation which is the change in position with regards to the sea or ocean level.
In this case, the harbour seal dives 35 metres below the ocean's surface and then it dives down an additional 12 metres
The new position will be;
= -35 - 12
= -47
This implies that it's 47 meters below the ocean surface.
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If tan 8 = 3 and 8 lies in the second quadrant. Use a Pythagorean identity to find
the exact value of cos 6.
The exact value of Cosθ is -1/√10 .
In the question ,
it is given that ,
the value of tanθ = 3/1 ,
that means the perpendicular = 3x
and the base of the triangle is = 1x,
So ,By Pythagoras Theorem
hypotnuse² = perpendicular² + base²
hypotnuse² = (3x)² + (1x)²
hypotnuse² = 9x² + 1x²
hypotnuse² = 10x²
hypotnuse = (√10)x
So , Cosθ = base / hypotnuse
Cosθ = 1x/(√10)x
Cosθ = 1/√10
Since , [tex]\theta[/tex] lies in the second quadrant ,
and in second quadrant Cos is negative ,
So , Cosθ = -1/√10
Therefore , The exact value of Cosθ is -1/√10 .
The given question is incomplete , the complete question is
If Tanθ = 3 , θ lies in the second quadrant. Use a Pythagorean identity to find the exact value of Cosθ ?
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