an urban planner collects data on how park trails are used by residents. the planner looks at two trails: one that winds through an urban area and another in a suburban park. the table shows the number of users who walk, jog, or bike the trail. activity walkjogbiketotal type of parkurban765823157 suburban1257642243 total20113465400 what proportion of suburban park users bike on the park trail? 0.1050 0.1728 0.6075 0.6461
0.1728 suburban park users bike on the park trail.
A mathematical comparison of two numbers is called a proportion.
It is an equation or statement used to depict that two ratios or fractions are equal.
According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.
The proportion (p) is calculated as:
p = bike the trail/number of residents
Given that,
The number of users who bike the trail = 42
The number of residents using the park trails = 243
Thus,
p = 42/243 = 0.1728
Therefore, 0.1728 suburban park users bike on the park trail.
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Which inequality represents all the solutions of 10(3x + 2) > 7(2x − 4)?
Answer:
Step-by-step explanation:
The inequality that represents all the solutions of 10(3x + 2) > 7(2x − 4) is 10(3x + 2) > 7(2x − 4). To solve this inequality, first we must combine like terms on each side of the inequality. On the left side, 10(3x + 2) becomes 30x + 20. On the right side, 7(2x − 4) becomes 14x − 28. After combining the like terms, the inequality becomes 30x + 20 > 14x − 28. Next, we must subtract 14x from both sides of the inequality. On the left side, 30x + 20 becomes 16x + 20. On the right side, 14x − 28 becomes −28. After subtracting 14x from both sides, the inequality becomes 16x + 20 > −28. Finally, we must subtract 20 from both sides of the inequality. On the left side, 16x + 20 becomes 16x. On the right side, −28 becomes −48. After subtracting 20 from both sides, the inequality becomes 16x > −48. To solve the inequality, we must divide both sides of the inequality by 16. On the left side, 16x becomes x. On the right side, −48 becomes −3. Therefore, the solution to the inequality is x > −3.
i am an advanced open water diver. i plan to do three dives. the first dive is a 25-meter dive for 22 minutes followed by a 52-minute surface interval. the second dive is to 20 meters for 27 minutes followed by a surface interval of 62 minutes. my third dive is to 18 meters. what would be my maximum allowable bottom time for this third dive?
use the 20m/70ft as the next depth and put 27 minutes for the criteria ABT. With the surface interval of 62 mins and G as the new pressure group. Use the 18m/60ft as the final depth, the final ABT will be approximately 34 mins.
If we use the general Dive Planning mode on the eRDPML and insert the option "No" for the Multilevel Dive. Furthermore, insert the option "Yes" for the First Dive. If we use 25m/80ft and click the enter button when the NDL appears. Also put 22 minutes for the criteria ABT. For the surface interval (i.e. 52 mins), E/D is the new pressure group.
i am an advanced open water diver. i plan to do three dives. the first dive is a 25-meter dive for 22 minutes followed by a 52-minute surface interval. the second dive is to 20 meters for 27 minutes followed by a surface interval of 62 minutes. my third dive is to 18 meters.
In addition, use the 20m/70ft as the next depth and put 27 minutes for the criteria ABT. With the surface interval of 62 mins and G as the new pressure group. Use the 18m/60ft as the final depth, the final ABT will be approximately 34 mins.
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give a thorough explanation of how both disparate treatment and disparate/adverse impact cases operate. a. what are they? b. how do you recognize them? c. how are they proven (what are the necessary elements)? d. what defenses are available to each? (describing how to prove such defenses.)
Discriminatory practises are those that have a disparate impact or treatment. When compared to differential treatment, which is deliberate discrimination, disparate impact is frequently referred to as inadvertent discrimination. Alternative words for this concept include adverse effect and adverse treatment.
How is disparate treatment defined?Disparate treatment discrimination, put simply, is when a company treats a current employee or potential employee improperly due to that person's race, religion, colour, sex, national origin, etc. This type of discrimination is usually the most apparent and visible.
What do disparate impact and treatment mean?Disparate treatment and disparate impact are two different things; disparate treatment is purposeful discrimination, whereas disparate impact is accidental.
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It is the morning of the community-wide yard sale that Joe has been anticipating. According to The Weather Channel there is a 10% probability of snow in his area today. Which of the following best describes the most likely way the probability was determined?
Historically, in this area, it has snowed 10% of the time on days with similar meteorological conditions as today.
It snows 10% of the time on this date each year.
Historically, in the United States, it has snowed 10% of the time on days with similar meteorological conditions as today.
Historically, it snows 10% of the days during this month.
Answer:
The answer is A.
Step-by-step explanation:
Lets prove the other ones wrong.
B) It is not based on a day to day prediction, rather days with simmilar conditions before the storm.
C) The United states is too broad an area.
D) It is not based on the ammount of days it snows per month, because on say December 15 it could be 60 degrees and on December 25 it could be 2 degrees. Also you could not make any predictions on a per-day basis then.
So the only clear answer is "Historically, in this area, it has snowed 10% of the time on days with similar meteorological conditions as today."
Anthony measured a line to be 16 inches long. If the actual length of the line is 15.6 inches, then what was the percent error of the measurement, to the nearest tenth of a percent?
Answer:
2.56% or 2.6% (to nearest 10th of a %)
Step-by-step explanation:
Percent error is determined by the expression
[(Measured - Actual)/Actual]*100%
[16" - 15.6"]/(15.6")*100% = 2.56%
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
A charitable organization in Wildgrove is hosting a black tie benefit. Yesterday, the organization sold 22 regular tickets and 74 VIP tickets, raising $10,274. Today, 48 regular tickets and 61 VIP tickets were sold, bringing in a total of $10,261. How much do the different ticket types cost?
The system of equations to describe the condition is,
22x + 74y = 10274
48x + 61y = 10261
What is an Equation ?An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.
For example, 3x+2y=0.
Types of equation
1. Linear Equation
2. Quadratic Equation
3. Cubic Equation
Given that,
Yesterday, the organization sold 22 regular tickets and 74 VIP tickets, raising $10,274
Today, 48 regular tickets and 61 VIP tickets were sold, bringing in a total of $10,261
Cost of regular ticket = ?
Cost of VIP ticket = ?
Suppose,
Cost of regular and VIP ticket are respectively, x & y
22x + 74y = 10274
48x + 61y = 10261
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Esmeralda draws a scale drawing of a rectangular community garden she helps manage. The scale factor from the drawing to the garden is 20. Based on the measurements of Esmeralda's drawing, what is the perimeter of the community garden
The perimeter of the rectangular community garden, after the size has been altered by a factor of 20, is 130 feet.
In two-dimensional shapes, a perimeter is defined to be a closed path that surrounds or outlines a shape. The perimeter of a circle or an ellipse though is called the circumference.
The perimeter P of a rectangle is equal to two times the length l plus two times the width w [P = 2(l + w)]. The length and the width of the rectangular community garden are consecutively 2,25 and 1 feet, and the scale factor is 20.
Therefore, the perimeter of the community garden is
P = 2(l + w) · 20
P = 2(2,25 + 1) · 20
P = 6.5 · 20
P = 130 feet.
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Answer:
130 feet
Step-by-step explanation:
Ekene was curious if sample range was an unbiased estimator of population range. She placed ping pong ballsnumbered from 0 to 32 in a drum and mixed them well. Note that the range of the population is 32.
She then took a random sample of 5 balls and calculated the range of the sample. She replaced the balls and
repeated this process for a total of 50 trials. Her results are summarized in the dotplot below, where each dot
represents the sample range from a sample of 5 balls.
.
ea
8
12
16
20
24
28
32
Sample range
Based on these results, does sample range appear to be a biased or unbiased estimator of population range?
Hint: The population range is 32.
The graph provided in the exercise must be used in this task, together with our statistical understanding, so that we can state: correct option is A
What is statistics?Statistics is a discipline of applied mathematics concerned with the gathering, description, analysis, and derivation of inferences from quantitative data. Calculus of differential and integrals, linear algebra, and probability theory are key mathematical concepts in statistics.
Statistics provide us the knowledge we need to understand how things operate. To conduct research, assess results, foster critical thinking, and make wise judgements, statistics are utilized. Almost any subject of study may benefit from the application of statistics to explore the causes, timing, and predictability of events.
The maximum points on the graph are at 21 and 31, and the sample ranges appear to be centered below 32, so we may claim so the option A is more appropriate.
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The complete question is as follows:
Please help me long division step by step !!!
Answer:
0.850
Step-by-step explanation:
The first step is to write the number in standard form. Then, you place the decimal point in the quotient above the decimal point of the dividend. Then, divide and bring down the other digits such as hundredths, thousandth, and ect.
Is 1 2 a half or half?
Answer: half
Step-by-step explanation:
An architect is designing a new building. He makes a model of the building such that the area of the rectangular base is 12x2 - 11x - 5 and the length is 3x + 1. The completed model will be in the shape of a rectangular prism where the volume is given by the polynomial 24x3 - 58x2 + 23x + 15. Determine the width and height of the model in terms of x. Fill in the values of m and b to complete the expressions. Type the width on the first line and the height on the second line.
Answer:
Step-by-step explanation:
width = 3x + 1
height = (24x^3 - 58x^2 + 23x + 15) / (12x^2 - 11x - 5)
part a: find the value of integral from 0 to 6 of f of x dx, or explain why the integral does not exist. (3 points) part b: find the value of integral from 4 to 9 of g of x dx. show the work that leads to your answer. (3 points) part c: find the value of integral from 4 to 9 of 2 times g of x minus f of x end quantity dx. show the work that leads to your answer. (4 points)
The anti-derivative of g(x) is G(x) = x3 + 3x2 and the anti-derivative of f(x) is F(x) = x2 + 2x.
Part a:
The value of the integral from 0 to 6 of f(x)dx is 16. This can be calculated using the Fundamental Theorem of Calculus (FTC) which states that the integral of a function is equal to the anti-derivative of that function evaluated at the upper limit minus the anti-derivative of that function evaluated at the lower limit. This can be written as:
∫f(x)dx = F(6) - F(0)
The anti-derivative of f(x) is F(x) = x2 + 2x, so the value of the integral is:
∫f(x)dx = (62 + 2*6) - (02 + 2*0) = 36 - 0 = 16.
Part b:
The value of the integral from 4 to 9 of g(x)dx is 20. This can be calculated using the FTC which states that the integral of a function is equal to the anti-derivative of that function evaluated at the upper limit minus the anti-derivative of that function evaluated at the lower limit. This can be written as:
∫g(x)dx = G(9) - G(4)
The anti-derivative of g(x) is G(x) = x3 + 3x2, so the value of the integral is:
∫g(x)dx = (93 + 3*9) - (43 + 3*4) = 729 - 168 = 20.
Part c:
The value of the integral from 4 to 9 of 2*g(x) - f(x)dx is 68. This can be calculated using the FTC which states that the integral of a function is equal to the anti-derivative of that function evaluated at the upper limit minus the anti-derivative of that function evaluated at the lower limit. This can be written as:
∫(2*g(x) - f(x))dx = 2*G(9) - F(9) - [2*G(4) - F(4)]
The anti-derivative of g(x) is G(x) = x3 + 3x2 and the anti-derivative of f(x) is F(x) = x2 + 2x, so the value of the integral is:
∫(2*g(x) - f(x))dx = (2*(93 + 3*9) - (92 + 2*9)) - [2*(43 + 3*4) - (42 + 2*4)]
= (1854 - 19) - (544 - 16) = 1735 - 528 = 68.
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what is the area of the shaded portion in the figure? If, it is a square and the lines are parts of a circle and the side of the square is 10 unit. Give answers in terms of pi.
A quarter-circle with a radius of 5 has the area: 25π/4, This is where the shaded area in the figure is located.
How do you calculate the size of a circle’s shaded sector?The area of a circle’s sector is calculated as /360o (r²), where r is the radius of the circle and is the angle of the sector.
The little, subtle distinctions between words are distinguished by shades of meaning. The terms “happy” and “ecstatic,” for example, both express emotion, but one is more powerful than the other. When students study shades of meaning, they learn how to effectively articulate how they feel or think. Shade is any gloomy place where sunshine or other strong light is obstructed. A shadow is a dark form that develops on a surface when an item blocks sunlight or other light.
The shaded area in the illustration is a quarter-circle with a radius equal to 10/2, or 5 units, or half the length of the square's side.
The following formula can be used to determine a circle's area:
A = πr²
If pi is present and r is the radius (approximately equal to 3.14).
A quarter-circle with a radius of 5 has the following area:
A = (π × 5²) / 4 = (π × 25) / 4 = 25π / 4
This is where the shaded area in the figure is located.
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Is multiplying by .5 the same as dividing by 2?
Answer:
yes because it's like multiplying my 1/2.
when you multiply a number with a fraction its the same as dividing.
Step-by-step explanation:
Example: 6/1 × 1/2 = 3
If x is the principal square root of 7, what must be true?
a. x = 72, and x is positive
b. x2 = 7, and x is negative
c. x2 = 7, and x is positive
d. x = 72, and x is negative
Answer:C
Step-by-step explanation:
x is [tex]\sqrt{7} \\[/tex]=2.645
which means 2.645²=7
what is the positive and negative to this question
The positive and negative roots of the given quadratic graph are; -2 and 4.
What are the roots of the quadratic graph?We want to find the positive and negative roots of the quadratic graphs. Now, the general form of quadratic equations is expressed as;
y = ax² + bx + c
Now, for a quadratic graph, the roots are defined as the value of x where the curve crosses the x-axis.
In this given graph, we can see that the y-intercept which is the point at which the curve crosses the y-axis is at y = -8. We also see that the points at which the graph crosses the x-axis are at x = -2 and x = 4. This clearly shows that one root is positive and one root is negative.
Therefore from the two x-intercept values above as well as the y-intercept we got and when considering the type of roots that are being looked for, we can easily conclude that the positive and negative roots are as gotten above.
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a circle with center $o$ has radius $8$ units and circle $p$ has radius $2$ units. the circles are externally tangent to each other at point $q$. segment $ts$ is the common external tangent to circle $o$ and circle $p$ at points $t$ and $s$, respectively. what is the length of segment $os$? express your answer in simplest radical form.
The length of segment is 24/√2
Given that,
Having radii of 2 and 4, respectively, and being externally tangent, are circles with centers O and P. The circle with the center at O has points A and B, while the circle with the center at P has points C and D, so that
AD and BC are frequent tangents to the outside of the circles.
To find : What is the hexagon's area?
ADPO and OPBC are congruent right trapezoids with legs 2 and 4 and with OP equal to 6. Draw an altitude from O to either DP or CP,
Creating a rectangle with width 2 and base x, and a right triangle with one leg 2, the hypotenuse 6, and the other aa Using the Pythagorean theorem, is equal to , and x is also equal to the height of the trapezoid. The area of the trapezoid is thus
length = 1/2 * (4+2).4√2 = 12√2
and the total length is two trapezoids, or 24/√2
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two fair dice are thrown together. what is the probability that the difference between the two outcomes is 3
The probability of getting a difference of 3 when two fair dice are thrown together is 1/9.
What is probability?Probability is the measure of how likely it is that a certain event will occur. It is expressed as a number between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that an event is certain. Probability can also be expressed as a percentage or fraction. Probability is used to describe the likelihood of an event occurring, and its calculation is based on the number of possible outcomes of the event.
This is due to the fact that there are 6 possible outcomes for each dice, and 36 possible outcomes when both dice are thrown. The only way to get a difference of 3 is for the first dice to show a 4 (1/6 probability) and the second dice to show a 1 (also 1/6 probability). This gives a combined probability of 1/6 x 1/6 = 1/36, which is 1/9.
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You are single and support your mom and dad. You earn $56,000 per year. The state income tax rate is 5. 5%. Find the annual tax withheld
Answer:
Step-by-step explanation:
To find the annual tax withheld, you would need to calculate the amount of income tax that you owe based on your income and the state income tax rate.
You earn $56,000 per year and the state income tax rate is 5.5%. To find the amount of tax you owe, you would multiply your income by the tax rate.
Tax Owed = Income x Tax Rate
Tax Owed = 56,000 x 0.055
Tax Owed = 3,080
So, the annual tax withheld from your salary would be $3,080.
Question
What equation is graphed in this figure?
Responses
y−4=−23(x+2)
y minus 4 equals negative fraction 2 over 3 end fraction left parenthesis x plus 2 right parenthesis
y+2=−32(x−2)
y plus 4 equals negative fraction 3 over 2 end fraction left parenthesis x minus 2 right parenthesis
y−3=32(x+1)
y minus 3 equals fraction 3 over 2 end fraction left parenthesis x plus 1 right parenthesis
y+1=−23(x−3)
y plus 1 equals negative fraction 2 over 3 end fraction left parenthesis x minus 3 right parenthesis
Number graph that ranges from negative five to five on the x and y axes. A line passes through begin ordered point zero comma one end ordered pair and begin ordered pair two comma negative two end ordered pair
The equation which is graphed in this figure (see attachment) is: C. y + 2 = −3/2(x − 2)
"y plus 4 equals negative fraction 3 over 2 end fraction left parenthesis x minus 2 right parenthesis."
How to determine the number of solutions?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y represents the data points.From the information provided about this graph, we have the following data points on its line:
Points on the x-axis = (0, 2).
Points on the y-axis = (1, -2).
At data point (0, 3), a linear equation of the first line can be calculated in slope-intercept form as follows:
y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
y - (-2) = (-2 - 1)/(2 - 0)(x - 2)
y + 2 = -3/2(x - 2)
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Circle O shown below has a radius of 3 inches. To the nearest tenth of an inch,
determine the length of the arc, x, subtended by an angle of 118°.
3 inches
O 0=118°
Answer: 6.2
Step-by-step explanation:
Using the arc length formula,
[tex]x=\frac{118}{360}(2)(3)(\pi) \approx 6.2[/tex]
Greta opens a savings account with $25. She saves $20 each week. The table represents her account balance. Complete parts a and b below.
Greta's Savings Account
Week
0
1
2
3
4
5
Money in Account
25
45
65
85
105
125
Question content area bottom
Part 1
a. Write a function that relates the amount of money in Greta's account m to the number of weeks w.
m=
Please help me omg just give me m=
Answer:
m(w) = $25+$20w
Step-by-step explanation:
Let m be the amount saved and w the weeks since the account was opened. Greta starts with $25, so m(0) = $25.
She saves $20 each week, w. m(1), the end of the first week, Greta would have $25 + $20 = $45. This continues for future weeks. We could write this as the follwoing:
m(w) = $25+$20w
Week 5, for example, would end with m(5) = $25+$20*(5)
m(5) = $125.
The curve above is the graph of a sinusoidal function.
The sine function that matches the graph is given as follows:
y = 3sin(π/5(x + 0.5))
How to define the sine function?The sine function for this problem is defined as follows:
y = Asin(B(x - C)) + D.
The parameters are described as follows:
A is the amplitude.The period is of 2π/B.C is the phase shift.D is the vertical shift.The function has a minimum value of -3 and a maximum value of 3, for a difference of 6, hence the amplitude is obtained as follows:
2A = 6
A = 3.
The function has no vertical shift, as it oscillates between -1 x 3 = -3 and 1 x 3 = 3.
The period is of 10 units, hence the coefficient B is given as follows:
2π/B = 10
10B = 2π
B = π/5.
Due to it's midline, the function was shifted 0.5 units left, hence the coefficient C is given as follows:
C = -0.5.
Hence the function is defined as follows:
y = 3sin(π/5(x + 0.5))
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g assume a quarterback completes 44% of his passes. what is the probability that the quarterback completes his first pass on his 4th attempt?
The probability that the quarterback completes his first pass on his 4th attempt is 0.44
The term probability in math is defined as the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes
Here we have given that assume a quarterback completes 44% of his passes.
Here we have to convert the percentage into decimal form, then we get,
=> 44/100
When we divide it, then we get the probability that the quarterback completes his first pass on his 4th attempt is written as,
=> 0.44
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Ava likes to go to her local hair salon because she is a premier member there. Her membership fee costs $80.00 and that membership allows her to get her hair done for $14.00 every visit. How much does it cost for 2,4,5 and 8 visits?
Check the picture below.
A circle is centered at O. The tangent to the circle at P is extended to Q. Line segment QS intersects the circle at R. Given that OS = 2, SR = RQ = 3, and PQ = 6, find the radius of the circle.
The radius of the circle will be [tex]\sqrt{22}[/tex].
In our question, we shall first extend QRS to the circle's point T.
According to the power of a point.
[tex]QP^{2}[/tex] = QR*QT
[tex]6^{2}[/tex] = 3*QT
QT = 12
Since QR = 3, RT = 9.
Since RS = 3, ST = 6.
Draw OT and OR, creating a triangle(TSO) and triangle(RSO).
OT and OR are radii, Let the radius be r.
Use the Law of Cosines on these two triangles.
OT2 = [tex]r^{2}[/tex] = [tex]6^{2}[/tex] + [tex]2^{2}[/tex] - 2*6*2*cos(TSO)
OR2 = r2 = [tex]3^{2}[/tex] + [tex]2^{2}[/tex] - 2*3*2*cos(RSO)
Since these are equal:
[tex]6^{2}[/tex] + [tex]2^{2}[/tex] - 2*6*2*cos(TSO) = [tex]3^{2}[/tex] + [tex]2^{2}[/tex] - 2*3*2*cos(RSO)
40 - 24*cos(TSO) = 13 - 12*cos(RSO)
Since angle(RSO) is supplementary to angle(TSO), cos(RSO) = - cos(TSO)
40 - 24*cos(TSO) = 13 + 12*cos(TSO)
27 = 36*cos(TSO)
0.75 = cos(TSO)
Since [tex]OT^{2}[/tex] = [tex]r^{2}[/tex] = [tex]6^{2}[/tex] + [tex]2^{2}[/tex] - 2*6*2*cos(TSO)
[tex]r^{2}[/tex] = 40 - 24*0.75
[tex]r^{2}[/tex] = 22
r = [tex]\sqrt{22}[/tex]
Hence the radius of the given circle will be [tex]\sqrt{22}[/tex].
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a book sold 34200 copies in its first month of release. suppose this represents 6.8% of the number of copies sold to date. how many copies have been sold to date.
On solving the provided question, we can say that equation let x =amount of copies sold = > 37,600is = .073 X x = 515,068.4932.
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.
let x =amount of copies sold to date.
37,600is = .073 X x
t x = 37,600 / .073
= 515,068.4932.
round to nearest 515,068.
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in how many ways can $5$ people be seated around a round table? (two seatings are considered the same if one is a rotation of the other.)
24 different seatings are possible when five people are to be seated around a circular table such that two seatings are considered the same if one is a rotation of the other.
What is fundamental counting principle?
A rule used to count the entire number of alternative outcomes in a circumstance is known as the fundamental counting principle. It asserts that if there are n ways to do something and then m ways to do something else, then there are nm methods to do both of these acts. The Fundamental Principle of Counting assists in determining how selection is made based on available options. The Fundamental Principle of Counting streamlines the method to choosing by taking into account all of the available options and their combinations in order to calculate the Probability.
Here,
Since there are 5 peoples,
let us fix the first person and remaining will be set according to that.
this can be arranged in,
(5-1)!=4!
=24 ways
When five persons are sitting around a circular table, 24 distinct seatings are available, with two seatings regarded the same if one is a rotation of the other.
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a spherical ball 8 inches in diameter is coated with a layer of ice of uniform thickness. if the ice melts at a rate of 10 cubic inches per min, how fast is the thickness of the ice decreasing when it is 2 inches thick? at the same rate of melting, how fast is the outer surface area of the ice changing?
The surface area of the ice is 16π square inches. At a rate of 10 cubic inches per min, the outer surface area of the ice is changing at 10/16π square inches per min.
To calculate this we have given:
The diameter of the spherical ball is 8 inches, so the radius of the ball is 8/2 = 4 inches.
The volume of a sphere is given by the formula (4/3)πr³, where r is the radius of the sphere.
The volume of the spherical ball with 2 inches of ice is (4/3)π(4³) = (4/3)π64 cubic inches.
The volume of the ice is the volume of the spherical ball with 2 inches of ice minus the volume of the spherical ball without the ice, which is (4/3)π(4³) - (4/3)π(4³ - (2³)) = (4/3)π64 - (4/3)π32 = 32π cubic inches
At a rate of 10 cubic inches per min, the thickness of the ice is decreasing at:
10 cubic inches / 32π cubic inches = 10/32π inches per min
To find the rate at which the outer surface area of the ice is changing, we can use the formula for the surface area of a sphere which is 4πr².
The surface area of the ice is 4π(4-2)³ = 4π4 = 16π square inches.
At a rate of 10 cubic inches per min, the outer surface area of the ice is changing at:
10 cubic inches / 16π square inches = 10/16π square inches per min
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