Answer:
y = x^2 + 8x + 19
Step-by-step explanation:
y=(x+4)²+3
First expand the parentheses:
y = x^2 + 4x + 4x + 16 + 3
y = x^2 + 8x + 19
6x−11>67 Solve the inequality by entering the correct answer in the box.
The solution to the inequality 6x - 11 > 67 is x > 13.
What is an inequality?An inequality in mathematics is a comparison between two numbers or expressions that shows one to be bigger than, less than, or not equal to the other. In algebra, inequalities are frequently used to illustrate the connections between variables. Symbols like (less than), > (greater than), (less than or equal to), (greater than or equal to), and can be used to denote an inequality (not equal to). For representing restrictions or limitations on a system, such as the maximum or minimum quantity of a resource that may be used, in real-world applications, inequalities are frequently employed.
The given inequality is:
6x−11>67
Adding 11 to both sides, we get:
6x > 78
Dividing both sides by 6, we get:
x > 13
Hence, the solution to the inequality 6x - 11 > 67 is x > 13.
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Write the quadratic equation whose roots are 2 and 4, and whose leading coefficient is 4.
(Use the letter to represent the variable.)
My shortest side stands alone
One of my sides is two less than five times it
The other side is two more than two times it.
Decode my numbers?
∠3 is in a triangle with ∠4 and ∠5. Write and solve an equation to find the m∠3
we must have made an incorrect assumption along the way. One possible mistake is to assume that the exterior angle of ∠3 is formed by extending the side opposite to ∠3.
What is Triangle ?
In geometry, a triangle is a 2-dimensional polygon that is formed by connecting three non-collinear points with straight lines.
To find the measure of ∠3, we can use the fact that the sum of angles in a triangle is 180 degrees. Therefore, we can write:
∠3 + ∠4 + ∠5 = 180
However, we don't know the measures of ∠4 and ∠5, so we need to use additional information from the problem to find them.
Since ∠3 is in the same triangle as ∠4 and ∠5, we can use the fact that the exterior angle of a triangle is equal to the sum of its remote interior angles. In other words:
∠4 + ∠5 = exterior angle of ∠3
We don't know the measure of the exterior angle of ∠3 either, but we can use the fact that it is supplementary to ∠3. Therefore:
exterior angle of ∠3 + ∠3 = 180
Combining these equations, we can solve for ∠3:
exterior angle of ∠3 + ∠3 = 180
∠4 + ∠5 = exterior angle of ∠3
∠3 + ∠4 + ∠5 = 180
Substituting ∠4 + ∠5 for the exterior angle of ∠3 in the first equation, we get:
∠3 + (∠4 + ∠5) = 180
∠3 + (exterior angle of ∠3) = 180
Simplifying, we get:
∠3 + (∠4 + ∠5) = 180
2∠3 + (∠4 + ∠5) = 180
2∠3 + exterior angle of ∠3 = 180
Now, we need to know the value of the exterior angle of ∠3. It is formed by extending one of the sides of the triangle containing ∠3, ∠4, and ∠5, and it is equal to the sum of the other two angles in the triangle. We don't have enough information to determine which side is extended, so we will assume that it is the one opposite to ∠3. In this case, we have:
exterior angle of ∠3 = ∠4 + ∠5
Substituting in the equation above, we get:
2∠3 + ∠4 + ∠5 = 180
Now, we can solve for ∠3:
2∠3 + ∠4 + ∠5 = 180
2∠3 = 180 - (∠4 + ∠5)
2∠3 = 180 - (exterior angle of ∠3)
2∠3 = 180 - (∠4 + ∠5)
2∠3 = 180 - (∠3 + ∠4 + ∠5)
2∠3 = 180 - 180
2∠3 = 0
∠3 = 0
However, this result doesn't make sense, because an angle can't have a measure of 0 degrees in a triangle.
Therefore, we must have made an incorrect assumption along the way. One possible mistake is to assume that the exterior angle of ∠3 is formed by extending the side opposite to ∠3.
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any one know the mixed number for this question?
4 1/4 - 5/6
Answer:
4 5/12
Step-by-step explanation:
Find the LCM of 6 and 4
This would be 12.
Multiply so that both denominators are 12.
For [tex]5\frac{1}{4}[/tex] multiply both the numerator and denominator by 3.
[tex]5\frac{3}{12}[/tex]
For [tex]\frac{5}{6}[/tex] multiply both the numerator and denominator by 2.
[tex]\frac{10}{12}[/tex]
Substitute the new values into the expression.
[tex]5\frac{3}{12} - \frac{10}{12}[/tex]
Solve
[tex]5\frac{3}{12} - \frac{10}{12} \\\\4\frac{15}{12} - \frac{10}{12} \\\\=4\frac{5}{12}[/tex]
The buying rate and selling rate of Australian dollar in a bank are Rs.132 and Rs 132.85 respectively.how much Australian dollar should be bought and sold by the bank to get Rs 7000 profit.find it.
Answer:
Let x be the amount of Australian dollars the bank buys and sells.
The bank earns a profit by selling the Australian dollars at a higher rate than it bought them.
Profit = Selling rate - Buying rate
Profit per unit = Selling rate - Buying rate = Rs.132.85 - Rs.132 = Rs.0.85
To earn a profit of Rs. 7000, the bank must sell x Australian dollars at a profit of Rs.0.85 per unit, so:
Profit = x * 0.85
x = Profit / 0.85
x = 7000 / 0.85
x = 8235.29
Therefore, the bank should buy and sell 8235.29 Australian dollars to earn a profit of Rs.7000.
Gary has a sheet of wrapping paper that is 96 cm long and 50 cm wide. He plans to use it to wrap a box with a surface area of 4,900 cm square. Does he have enough paper? If not, how much more paper does he need?
Answer: No it would not be enough , he will need 100 cm more of paper.
Step-by-step explanation:
First ,
The area of paper is
= 96 × 50
= 4800
How much more paper he needs = 4900 - 4800
= 100 cm
I hope this helps
Walter bought 2,000 cubic millimeters of clay for his sculpture. Now that he has molded the cone and the sphere, he wants to use the rest for decorations. How much clay does he have left for decorations?
Answer:
Step-by-step explanation:
Let's first find the total volume of clay used for the cone and the sphere:
For the cone, we need the formula for the volume of a cone: Vcone = (1/3)πr^2h, where r is the radius of the base and h is the height.
Let's assume the cone has a radius of 5 millimeters and a height of 10 millimeters. Then, the volume of the cone is:
Vcone = (1/3)π(5^2)(10) = (1/3)π(250) = 83.33 cubic millimeters (rounded to two decimal places)
For the sphere, we need the formula for the volume of a sphere: Vsphere = (4/3)πr^3, where r is the radius.
Let's assume the sphere has a radius of 4 millimeters. Then, the volume of the sphere is:
Vsphere = (4/3)π(4^3) = (4/3)π(64) = 268.08 cubic millimeters (rounded to two decimal places)
The total volume of clay used for the cone and the sphere is:
Vcone + Vsphere = 83.33 + 268.08 = 351.41 cubic millimeters (rounded to two decimal places)
To find the amount of clay left for decorations, we can subtract this volume from the original amount of clay:
2000 - 351.41 = 1648.59 cubic millimeters (rounded to two decimal places)
Therefore, Walter has 1648.59 cubic millimeters of clay left for decorations.
How do I find the area of the shape
Answer:
To find the area of a shape, you need to know the formula for the specific shape you are dealing with. Here are some common shapes and their corresponding area formulas:
Rectangle: A = l x w (where A is the area, l is the length, and w is the width)Square: A = s^2 (where A is the area and s is the length of one side)Triangle: A = (1/2)bh (where A is the area, b is the base of the triangle, and h is the height)Circle: A = πr^2 (where A is the area and r is the radius)Once you have identified the shape and the formula for its area, plug in the appropriate values and calculate the area. Make sure to label your answer with the correct units (e.g. square inches, square feet, etc.).
Pregunta 4
Lea la pregunta y escriba su respuesta en el cuadro proporcionado. Su respuesta se guardar
Considere la función:
y = x² + 2X +3
¿Cuáles son las coordenadas del vértice de la función dada?
The vertex form of the given function reveals that a = 1, h = -1, and k = 2 are true. Hence, the vertex's coordinates are (-1, 2).
what is function ?A function is a mathematical rule that gives each input value a distinct output value. A function, then, is a relationship between two sets of numbers in which each element of the first set (referred to as the domain) corresponds to exactly one element of the second set (called the range). An equation or formula that describes how input values are changed into output values defines a function, which is typically represented by a letter, such as f. For instance, the rule represented by the function f(x) = x2 produces the value x squared from any input value x.
given
We must finish the square and rewrite the function in vertex form in order to determine the vertex's coordinates for the provided function, y = x2 + 2x + 3.
y = x² + 2x + 3
y = (x + 1)² - 1 + 3 (completing the square)
y = (x + 1)² + 2 (simplifying) (simplifying)
A quadratic function's vertex form is denoted by y = a(x - h)2 + k, where (h,k) denotes the vertex's coordinates.
The vertex form of the given function reveals that a = 1, h = -1, and k = 2 are true. Hence, the vertex's coordinates are (-1, 2).
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The complete question :- Read the question and write your answer in the box provided. Your answer will be saved
Consider the function:
y = x² + 2X +3
What are the coordinates of the vertex of the given function?
a rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. find the dimensions of the garden that minimize the cost of materials.
The dimensions of the garden that minimize the cost of materials are L = 15√15 feet and W = 5√15 feet.
To minimize the cost of materials for the given rectangular garden, we need to find the dimensions that minimize the total cost of the brick wall and the fence.
Let's assume the length of the garden is L and the width is W. Then, the area of the garden is given by LW = 75. The garden is to be surrounded on three sides by the brick wall, so the total length of the wall required is 2L + W.
The cost of the brick wall is $10 per foot, so the cost of the wall is 10(2L + W). The cost of the fence on the remaining side is $5 per foot, so the cost of the fence is 5W.
Hence, the total cost of materials C is given by:
C = 10(2L + W) + 5W
Simplifying this equation, we get:
C = 20L + 15W
Using the area formula LW = 75, we can solve for one of the variables in terms of the other. For example, we can solve for W as follows:
W = 75/L
Substituting this into the equation for C, we get:
C = 20L + 15(75/L)
To minimize C, we can take the derivative of C with respect to L, set it equal to zero, and solve for L:
dC/dL = 20 - 1125/L^2 = 0
Solving for L, we get:
L = 15√15
Substituting this value of L back into the equation for W, we get:
W = 5√15
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If a boat weighs 50,000 pounds, how many tons will it weigh?
Answer: 25 tons
Step-by-step explanation:
there are 2000 lb in one ton
50000lb * (1 ton/2000lb) = 25 tons
rewrite each exponent as a single number to make this equation easier to work with. What are the values of 16 2 and 12 2 ?
Exponents are a mathematical notation used to represent repeated multiplication and are a fundamental tool in various fields. The values of [tex]16^2[/tex] and [tex]12^2[/tex] are [tex]16^2[/tex] = 16 x 16 = 256, [tex]12^2[/tex] = 12 x 12 = 144
To rewrite each exponent as a single number, we can evaluate them using multiplication. For example, [tex]16^2[/tex] can be rewritten as 16 x 16, which equals 256. Similarly, [tex]12^2[/tex] can be rewritten as 12 x 12, which equals 144.
The exponent is a small number written above the base, which indicates the number of times the base should be multiplied by itself.
For example, the exponent 3 written above the base 2 means that we should multiply 2 by itself three times: 2 x 2 x 2 = 8. So, [tex]2^3[/tex] is equal to 8.
So, the values of [tex]16^2[/tex] and [tex]12^2[/tex] are:
[tex]16^2[/tex] = 16 x 16 = 256
[tex]12^2[/tex] = 12 x 12 = 144
By evaluating the exponents as single numbers, we can simplify the expressions they appear in and make them easier to work with.
They are used to represent large and small numbers in a compact form, and they simplify calculations by allowing us to perform repeated multiplication more efficiently.
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Trisha is 3 times as old as Lilly, and Lilly is twice as old as Emma. If the sum of the three girls’ ages is 27, how old is Emma?
45,52,17,63,57,42,54,58 outlier
To identify an outlier in a set of numbers, we first need to determine the central tendency of the data, such as the mean or median. One common method for identifying outliers is to use the interquartile range (IQR).
To do this, we first need to find the median of the data set:
45, 52, 17, 63, 57, 42, 54, 58
Arranging them in ascending order:
17, 42, 45, 52, 54, 57, 58, 63
The median is the middle value, which is 54.
Next, we need to find the IQR. The IQR is the range between the first and third quartiles of the data. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half of the data.
To find Q1 and Q3, we split the data into two halves:
Lower half: 17, 42, 45, 52
Upper half: 54, 57, 58, 63
Q1 is the median of the lower half, which is (42 + 45)/2 = 43.5.
Q3 is the median of the upper half, which is (57 + 58)/2 = 57.5.
Therefore, the IQR is 57.5 - 43.5 = 14.
Finally, we can identify outliers as any data point that falls outside the range of 1.5 times the IQR above Q3 or below Q1.
The upper limit is Q3 + 1.5(IQR) = 57.5 + 1.5(14) = 78.5.
The lower limit is Q1 - 1.5(IQR) = 43.5 - 1.5(14) = 22.5.
The only number in the given set that falls outside this range is 17, which is less than the lower limit. Therefore, 17 is the outlier in this data set.
Answer:
17
Step-by-step explanation:
In the set of numbers: 45, 52, 17, 63, 57, 42, 54, 58, the outlier is the number 17.
An outlier is a data point that is significantly different from other data points in the set. In this case, 17 is much smaller than the other numbers in the set and is considered an outlier.
For questions 7-10, consider the trinomial 16x² + 16x-5.
7.
List all the factor pairs for -80.
8. Find the factor pair for -80 that add to the sum of 16.
9. Use question 8 to write the expanded form of 16x² + 16x - 5 that can be u
factor the expression by grouping.
10. Factor the expression 16x² + 16x-5 by grouping.
Therefore, the factored form of the expression is:(2x - 1)(8x + 5)
Factor pairs of80 and 1, 40 and 2, 20 and 5, 16 and 8, and 10
In question 8 This requirement is met by the pair -8 and 24.
In question9 16x² - 8x + 24x - 5
7. We can list all the pairs of numbers that multiply to -80 in order to identify all the factor pairs for this number. These are a few of the pairs:
80 and 1, 40 and 2, 20 and 5, 16 and 8, and 10
Define highest common factor?The biggest integer that divides each of two or more numbers without producing a remainder is known as the greatest common factor (GCF).
For instance, 6 is the highest number that divides both 12 and 18 without producing a residual, making it the GCF of 12 and 18.
The highest common factor (HCF) and greatest common divisor (GCD) are other names for the GCF (HCF).
8. We can seek for two values in the list from question 7 that add up to 16 in order to get the factor pair for -80 that adds to the sum of 16. This requirement is met by the pair -8 and 24.
9. Applying the answer to question 8, we can write 16x2 + 16x - 5 in its expanded form as:
16x² - 8x + 24x - 5
The terms can then be categorised as follows:
(16x² - 8x) + (24x - 5) (24x - 5)
When we take the biggest thing in common between each group, we get:8x(2x - 1) + 5(4x - 1) (4x - 1)
10. As a result, grouping can factor the enlarged form of 16x2 + 16x - 5 as follows:
8x(2x - 1) + 5(4x - 1) (4x - 1)
10. Using the enlarged form from answer 9, we may factor the formula 16x2 + 16x - 5 by grouping:
8x(2x - 1) + 5(4x - 1) (4x - 1)
We can factor it out because we can see that the common factor for both terms is (2x - 1):
(2x - 1)(8x + 5)
As a result, the expression's factored form is as follows:
(2x - 1)(8x + 5)
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don’t mind the work there. i need help solving the question please
Step-by-step explanation:
sin = -3/5 we are told this is between pi/2 and 3pi/2
so Quadrant II and III
of these two quadrants
sin is only negative in Q III where cos is also negative
TRIG IDENTITY: sin^2 + cos^2 = 1 will show cos = - 4/5
TRIG IDENTITY : Sin (2Φ) = 2 sinΦ cosΦ
= 2 (-3/5) (-4/5) = 24/25 = .960
Evaluate 4(x - 3) + 5x - x² for x = 3.
O A. -6
OB. -2
O C. 6
OD. 2
demarcus wants to know if the day of the week or the time of the day impacts exam scores more. he sets up a between-subjects (independent groups) factorial design and he has 6 conditions (3 x 2 design). he needs to have 50 people in each condition. how many total people would he need to recruit for his study?
Demarcus would need to recruit a total of 300 people for his study.
To determine this, you can follow these steps:
1. Identify the number of conditions in the factorial design. In this case, it is a 3 x 2 design, which means there are 6
conditions (3 levels of one factor multiplied by 2 levels of the other factor).
2. Determine the number of participants needed for each condition. Demarcus needs 50 people in each condition.
3. Multiply the number of conditions by the number of participants per condition. In this case, 6 conditions x 50
participants per condition = 300 total participants.
So, Demarcus needs to recruit 300 people for his between-subjects factorial design study.
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10 POINTS! PLEASE HELP! URGENT! ONLY CORRECT ANSWERS!
Answer:1.98
Step-by-step explanation: I know
Answer:
1 ft squared
Step-by-step explanation:
area = base x height / 2
2/3 x 3 = 2/2=1
a fair coin is flipped four times. What is the probability it lands exactly once. show a full solution
Answer:
1/4
Step-by-step explanation:
If the radius of a circle is 6 mm, the area is
6 mm²
127 mm2
367 mm2
By answering the presented question, we may conclude that area of the circle is = 113.1 mm².
What is circle?A circle seems to be a two-dimensional component that is defined as the collection of all places in a jet that are equidistant from the hub. A circle is typically shown with a capital "O" for the centre and a lower portion "r" for the radius, which represents the distance from the origin to any point on the circle. The formula 2r gives the girth (the distance from the centre of the circle), where (pi) is a proportionality constant about equal to 3.14159. The formula r2 computes the circumference of a circle, which relates to the amount of space inside the circle.
area of a circle A = πr²,
A = π(6 mm)² = π(36 mm²) ≈ 113.1 mm²
area = 113.1 mm².
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The surface of a cylinder is represented [tex]A = 2\pi r^{2} + 2\pi rh[/tex], where r is the radius of the cylinder and h is its height. Factor the right side of the formula.
Therefore, the factored form of the right side of the formula is 2πr(r + h).
What is cylinder?A cylinder is a three-dimensional geometric shape that consists of a circular base and a curved surface that connects the base's circumference to a parallel circle at the other end of the shape. It is a type of prism with circular bases. The two circular bases of the cylinder are parallel and congruent, meaning that they have the same size and shape. The curved surface of the cylinder is composed of all the points that are equidistant from both bases. The cylinder is a common shape in real-life objects, such as cans, pipes, and drinking glasses. Its volume and surface area formulas are frequently used in geometry and applied mathematics.
Here,
Starting with the given formula:
A = 2πr² + 2πrh
We can factor out a common factor of 2πr:
A = 2πr(r + h)
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translate the word problem below into an equation, then solve. The medieval spice merchant wants to blend some garlic powder, which costs 38 cents per pound, with 7 pounds of ground pepper, which costs 44 cents per pound, to create a big batch of medieval smelling salt worth exactly 40 cents per pound. How many pounds of the garlic powder should he use?
Answer:
Step-by-step explanation:
x = pounds of gralic powder
x + 7 = pounds of medieval smelling salt
(.38)(x) + (7)(.44) = (x + 7)(.40)
.38x + 3.08 = .40x + 2.80
.38x - .38x + 3.08 = .40x - .38x + 2.80
3.08 = .02x + 2.80
3.08 - 2.80 = .02x + 2.80 -2.80
.28 = .02x
.28/.02 = .02x/.02
14 = x
use 14 pounds of garlis powder
The list shows the amount of flour in pounds used by a bakery each day for 15 16, 17, 18, 19, 20, 23, 24, 29, 30, 31, 32, 32, 32, 32, 35 Which box plot best displays a summary of these data?
Option C's box-plot best depicts a data representing the list that shows the bakery using the amount of flour in pounds each day for data 15, 16, 17, 18, 19, 20, 23, 24, 29, 30, 31, 32, 32, 32, 32, and 35.
What exactly is a box plot?
The data in the five-number summary is represented by a box-whisker plot, also known as a box plot. This graph primarily displays the minimum observation, first-quartile, median, third-quartile, and maximum observation. A box is used to depict the first to third quartiles in a box plot. At the median, a vertical line segment goes through the box. The dots or line end points represent the minimum and maximum values or observations. The whisker represents the median of the data.
The following information is provided: 15,16,17,18,19,20,23,24,29,30,31,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32
1)Observation minimum=15
2)The maximum number of observations is 35.
3)The data's median:
Because the number of observations is even, the median will be the average.N=total number of observations
The median is the average or mean of the eighth and ninth observations.
= average of 24 and 29
=
=26.5
Median=26.5
4)First Quartile (Q1): the mean of the observations prior to the median.
=average of the first to seventh observations
= average of 15, 16, 17, 18, 19, 20, and 23
=
=18.28
5)Third quartile (Q3): Observational mean after median
=mean of the tenth to sixteenth observations
=average of 30, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32
=
=32
Q3-Q1 interquartile range (IQR)
=32-18.28
=13.72
Option-C contains a box plot that matches all of the values.
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an airplane door is 19 feet off the ground and the ramp has a 31 angle of elevation. What is the length of the ramp (y)?
The length of the ramp (y) is C. 36.9 feet
We can use trigonometry to find out the answer. Here we have given the opposite side length and the angle of elevation. Hence we can use the trigonometric ratio of tangent.
Let's see the steps for solving the problem.
Let AB be the height of the airplane door from the ground, and BC be the ramp's length. Also, let angle BAC be the angle of elevation of the ramp from the ground.
So,angle BAC = 31°andAB = 19 feet.
To find the length of the ramp (BC), we can use the tangent ratio of the angle of elevation.
tan θ = perpendicular/base
Here, the perpendicular is AB and the base is BC.tan 31° = AB/BC
Now, substitute the given values in the equation.
tan 31° = 19/BC
1/√3 = 19/BC
√3 = 19/BC
BC = 19/√3 feet [dividing both sides by √3]
We need to simplify the above answer.
We can multiply and divide the numerator by √3.BC = 19/√3 * √3/√3BC = 19√3/3
Therefore, the length of the ramp is BC = 19√3/3 feet.
Hence, the correct option is (C) 36.9 feet.
The question was incomplete, Find the full content below:
An airplane door is 19 feet off the ground and the ramp has a 31 angle of elevation. What is the length of the ramp (y)?
A. 31.6 feet
B. 22.2 feet
C. 36.9 feet
D. 9.8 feet
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please help 15 points
The numbers in this chart represent the number of minutes a group of five high school students spent talking on their cell phones one day.
Which of the following statements is true about this data?
A. The data set has two modes.
B. The data set has no mean.
C. The median is 41.
D. The mode and median are equal.
Answer:
It's probably C.
Step-by-step explanation:
41 is in the middle.
So technically speaking... It's the median?
It's definitely not A, that's for sure.
Edit: I was extremely wrong.
a motorboat travels -3.4 miles per hour for o.75 hours how far did it go
The negative sign indicates that the boat traveled in the opposite direction of its intended destination, and it traveled a distance of 2.55 miles.
The speed of the motorboat is given as -3.4 miles per hour, which means that the boat is moving in the opposite direction of its intended destination. If we assume that the boat is moving at a constant speed of -3.4 miles per hour for 0.75 hours, we can use the formula:
distance = speed x time
where distance is the distance traveled, speed is the speed of the boat, and time is the time for which the boat travels at that speed.
Plugging in the given values, we get:
distance = -3.4 miles/hour x 0.75 hour
distance = -2.55 miles
The negative sign indicates that the boat traveled in the opposite direction of its intended destination, and it traveled a distance of 2.55 miles.
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a motorboat travels -3.4 miles per hour for o.75 hours how far did it go?
what is The square with A = 225 in^2 equal to
The answer is s=√225 in. The area of a square is A = s², meaning that the area of the square is equal to the length of one side of the square squared.
What is square?A four-sided polygon with all sides equal in length. The interior angles of a square are right angles, and the opposite sides are parallel. The four sides of a square meet at its four vertices.
In this case, s is the side length.
To find the side length, we must take the square root of 225 in², which is equal to 15 in.
Therefore, the side length of the square is s=√225 in.
This is because taking the square root of a number is the same as finding the number whose square is equal to the given number.
In this case, the number whose square is equal to 225 in² is 15 in.
Hence, the side length of the square is s=√225 in.
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for a cylinder with a surface area of 90 , what is the maximum volume that it can have? round your answer to the nearest 4 decimal places.
The maximum volume that a cylinder with a surface area of 90 can have is approximately 27.05 cubic units
To solve this problem, we need to use the formulas for the surface area and volume of a cylinder
Surface area = 2πr^2 + 2πrh
Volume = πr^2h
where r is the radius of the base of the cylinder, h is the height of the cylinder, and π is approximately equal to 3.14159.
We want to find the maximum volume that a cylinder with a surface area of 90 can have. Let's first solve the surface area formula for h
90 = 2πr^2 + 2πrh
45 = πr^2 + πrh
45/π = r^2 + rh/π
Next, we can solve the volume formula for h
V = πr^2h
h = V/(πr^2)
Now we substitute this expression for h into the surface area formula
45/π = r^2 + r(V/(πr^2))
45/π = r^2 + V/r
To find the maximum volume, we need to find the value of V that maximizes this expression. We can do this by taking the derivative with respect to r and setting it equal to zero
d/dx (r^2 + V/r) = 2r - V/r^2 = 0
2r = V/r^2
r^3 = V/2
Now we can substitute this expression for V into the surface area formula to find the corresponding value of r
45/π = r^2 + r(V/(πr^2))
45/π = r^2 + r/2r^2
45/π = r^2 + 1/(2r)
45/π - 1/(2r) = r^2
r^2 = (45/π - 1/(2r))
r^4 = 45r/(π) - 1/4
We can solve this quartic equation for r using numerical methods, such as the Newton-Raphson method. Alternatively, we can use trial and error to approximate the value of r that satisfies this equation
V ≈ 27.05 cubic units
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