The amount deposited in the Patricia retirement savings plan are:
Amount deposited in the twentieth year is $1150.
Total amount deposited in time span of twenty years is $13,500.
Amount deposited by Patricia in the first year ' a₁ ' = $200
Difference between the amount deposited in the subsequent year is
'd' = $50
Amount deposited during 20th year
Let 'n' be the 20th year
The above situation represents it is arithmetic progression problem
nth term of A.P. is :
aₙ = a₁ + ( n - 1 ) d
Substitute the value we get,
a₂₀ = 200 + ( 20 - 1 ) 50
⇒ a₂₀ = 200 + 19 × 50
⇒ a₂₀ = 200 + 950
⇒ a₂₀ = $1150
Total amount deposited in twenty years
S₂₀ = (n/2) [ 2 a₁+ ( n - 1 )d ]
Substitute the value we get,
⇒S₂₀ = ( 20 /2 ) [ 2(200 ) + 19 × 50 ]
⇒ S₂₀ = 10 [ 400 + 950 ]
⇒S₂₀ = $13,500
Therefore, the amount deposited in the twentieth year is $1150 and total amount deposited in twenty years is $13,500.
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3. Consider two dots, X and Y, 7cm apart horizontally.
a. Draw the locus of points equidistant from X and Y.
b. A third dot Z is 6cm below X. Mark clearly the point equidistant from X, Y, and Z on your diagram in part a.
According to the information, the center line would be 3.5 cm from points X and Y.
How to graph the instructions described?To graph the instructions described we must perform the following procedure:
First of all we must carefully read the instructions and identify what the first step is. In this case we must draw a horizontal line 7 cm long.
Secondly, we must draw a line at the point equidistant from the points X and Z, that is, halfway between these two points. If we take into account that the first line measures 7 cm, this point will be 3.5 cm from each point.
Thirdly, we must give a length of 6 cm to this central line and it must also be equidistant. Therefore, we must give it 3 cm upwards and 3 cm downwards in length so that it is equidistant.
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principal $3200 rate 2% time 4 years
answer
Interest on amount after 4 years = $256
What is simple interest?Simple interest (S.I.) is the method of calculating the interest amount for a particular principal amount of money at some rate of interest for a given time.
Given,
Principal = $3200
Rate = 2% = 0.02
Time = 4 years
Simple Interest = Principal × rate × time
= 3200 × 0.02 × 4
= $256
Hence $256 is simple interest on Principal amount after 4 years.
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Look at pic for info
36) name four pairs of vertical angles
37) name two pairs of alternate interior angles
38) name all angles congruent to angle 1
39) name all angels congruent to angle 2
40) name two pairs of supplementary angles
41) name two pairs of corresponding angles
Based on the given angles:
4 pairs of vertical angles: ∠1 and ∠4; ∠2 and ∠3; ∠5 and ∠8; ∠6 and ∠72 pairs of alternate interior angles: ∠3 and ∠6; ∠4 and ∠5All angles congruent to angle ∠1: ∠4, ∠5, ∠8All angles congruent to angle ∠2: ∠3, ∠6, ∠72 pairs of supplementary angles: ∠1 and ∠2; ∠1 and ∠32 pairs of corresponding angles: ∠1 and ∠5; ∠2 and ∠6From the given lines and angles, we can identify:
Vertical angles are pair of angles with no adjacent line but share one meeting point and have congruent angles. Four pairs of vertical angles are:
∠1 and ∠4∠2 and ∠3∠5 and ∠8∠6 and ∠7Alternate interior angles are pair of angles within the parallel line (inside) that have congruent angles. Each pair cross the transversal line "t". Two pairs of alternate interior angles are:
∠3 and ∠6∠4 and ∠5Cogruent angles might come from several types of relationships, such as:
Vertical anglesAlternate interior anglesAlternate exterior angles Corresponding anglesAll angles congruent to angle ∠1 are:
Vertical angle: ∠4Corresponding angle: ∠5External alternate angle: ∠8All angels congruent to angle ∠2 are:
Vertical angle: ∠3Corresponding angle: ∠6Alternate external angle: ∠7Supplementary angles are a pair of angles whose its sum is 180°. Supplementary angles might formed by several types of angles relationship, such as:
Consecutive Interior anglesAdjacent anglesPairs of supplementary angles are:
Consecutive interior angles:
∠3 and ∠5∠4 and ∠6Adjacent angles:
∠1 and ∠2∠1 and ∠3∠2 and ∠4∠3 and ∠4∠5 and ∠6∠5 and ∠7∠6 and ∠8∠7 and ∠8Corresponding angles are a pair of unadjacent angles but have congruent angles because they face the same direction. Pairs of corresponding angles:
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The ratio between the areas of equilateral triangles $\triangle BXY$ and $\triangle BAC$ is $1:4.$ If $AC=6,$ what is $XY?$
Answer: b y c a x b is the answer
:
Find 1 sequences that are geometric and then write the recursive form and the explicit equation. I
The geometric sequence is given as follows:
1, 2, 4, 6, 8, ...
The recursive formula is given as follows:
f(n) = 2 x f(n - 1).f(1) = 1.The explicit formula is given as follows:
[tex]f(n) = 2^{n - 1}[/tex]
What is a geometric sequence?A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q, meaning that each term is obtained by the multiplication of the previous term by the common ratio q.
The explicit formula of a geometric sequence is given by:
[tex]f(n) = a_1q^{n-1}[/tex]
In which [tex]a_1[/tex] is the first term.
For the recursive formula, we have that:
f(n) = q x f(n - 1).
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It takes Jacob 30 minutes to ride his bike to school at 6 mph. Following the same route, how long would it take him to walk to school at 1.5 mph. (d = st)
The time it would take him to walk to the school at 1.5 mph is given as follows:
2 hours.
What is the relation between velocity, distance and time?Velocity is distance divided by time, hence the following equation is built to model the relationship between these variables:
v = d/t.
The parameters for this problem are given as follows:
v = 6, t = 0.5 hours.
Hence the distance is of:
d = vt
d = 6 x 0.5
d = 3 miles.
At a velocity of 1.5 miles per hour, the time would be given as follows:
1.5 = 3/t
1.5t = 3
t = 2 hours.
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A segment is connected to two sides of a triangle so that itnis parallel to the third side. Solve for x. A. 5 B. 4 C. 6 D. 9
The value of x in the angle is 56 degrees
How to detrmine the value of xFrom the question, we have the following parameters that can be used in our computation:
The figure
On the figure, we have teh following equation
64 + x + 60 = 130
The above is by the sum of angle theorem
So, we have
124 + x = 130
Evaluate the like terms
x = 6
Hence, the value of x is 6
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The following dot plots show the number of hours of sleep employees got before a major sale at two stores. Each dot represents a different employee.
Complete the comparison of the typical hours of sleep.
In general, the employees at
got more sleep, with
of sleep per employee.
In general, the employees at Store 1 got more sleep, with an average of 8.5 hours of sleep per employee, compared to Store 2's average of 7.5 hours.
The dot plots for the two stores show that Store 1 had employees who got between 5 and 10 hours of sleep, whereas Store 2 had employees who got between 4 and 8 hours of sleep. The median for Store 1 was 8.5 hours of sleep, and the median for Store 2 was 7.5 hours of sleep. This indicates that, in general, the employees at Store 1 got more sleep than the employees at Store 2. The dot plots also show that there were more employees at Store 1 with 8 hours of sleep than Store 2, and fewer employees at Store 1 with 6 hours of sleep than Store 2. This suggests that Store 1 had more employees getting 8 hours of sleep, and fewer employees getting 6 hours of sleep, than Store 2. This can be further seen by the fact that the average amount of sleep for employees at Store 1 was 8.5 hours, while the average amount of sleep for employees at Store 2 was 7.5 hours. Therefore, in general, the employees at Store 1 got more sleep than the employees at Store 2, with an average of 8.5 hours of sleep per employee, compared to Store 2's average of 7.5 hours.
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Please help me !!!!
This figure shows circle O with chords PM and LN
mPL=10 degrees
mLM= 14 degrees
mLN=24 degrees
PM= 15in.
What is LN
In this question, the length of the chord LN will be 15 inches long. It will be equal to PM.
Given,
At the circle's Centre, the Arc PL subtends an angle of 10 degrees.
At the circle's Centre, the Arc LM subtends an angle of 14 degrees.
At the circle's Centre, the Arc LN subtends a 24 degree angle.
The PM chord measures 15 inches in length.
What are the circle's chord theorems?
Considering the attached figure now,
The angle that the Arc PM creates at the Centre is equal to the sum of the angles that the Arc PL and Arc LM make there.
So,
At the middle, Arc PM will create the following angle: 10+14=24 degree.
We are aware that if the chords that make up the arc are equal, then the angle that it subtends in the Centre of the circle will likewise be equal.
Since, PM= 15 inches, therefore LN will also be 15 inches.
LN is therefore 15 inches as well.
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A researcher recorded the height of all 217 adult members of a tribe. The average was 5.49 feet.
What term best describes the analysis conducted by the researcher?
The term that best describes the analysis conducted by the researcher is; Observational Study
How to identify the system of research?We are given the parameters as;
Number of all adult members of a tribe = 217
Average height of the adult members = 5.49 ft
Now, there are different methods of research analysis such as experiment
randomized trial
double blind study
observational study
Now, in this case we are only told what the researcher has observed from his analysis and an observational study is used to answer a research question based purely on what the researcher observes.
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or
Watch a video
Find the cosine of ∠U.
28
45
53
U
V
T
Simplify your answer and write it as a proper fraction, improper fraction, or whole number.
Cosine of ∠U is 28/53 in proper fraction, 28/53 in improper fraction, 0.52609 in whole number.
The cosine of an angle in a right triangle can be found using the ratio of the adjacent side to the hypotenuse. In this triangle, the adjacent side to ∠U is 28 and the hypotenuse is 53.
cos(∠U) = adjacent side ÷ hypotenuse = 28 ÷ 53 = 28/53
So, the cosine of ∠U is 28/53, written as a proper fraction.
Now let's convert cosine of ∠U is 28/53 in improper fraction:
The cosine of ∠U, which is 28/53, can be expressed as an improper fraction by dividing both the numerator and denominator by their greatest common factor.
gcd(28, 53) = 1, so the fraction cannot be further simplified.
Therefore, the cosine of ∠U, expressed as an improper fraction, is 28/53.
Now let's convert cosine of ∠U is 28/53 in whole number:
The cosine of ∠U, which is 28/53, can be expressed as a whole number by dividing the numerator by the denominator.
28 ÷ 53 = 0.526087
So, the cosine of ∠U expressed as a whole number is approximately 0.52609
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_____ The given question is not correct, so correct Question is:
Find the cosine of ∠U in Triangle UTV with side:
UV=28
VT=45
TU=53
Simplify your answer and write it as a proper fraction, improper fraction, or whole number.
Can someone help me with this. Under lines are: ║ p and q. The statement is: ∠1 and ∠3 are vertical to ∠s. The reasons are: given, vertical angels theorem, transitive property, and def. of vertical ∠s.
The proof for the the lines p and q are parallel is given below.
What are Alternate Exterior Angles?Alternate exterior angles are the angles formed on the two opposite sides of the transversal which are on the outer side of the parallel lines.
Given that,
∠1 ≅ ∠2 or ∠2 ≅ ∠1.
Now ∠1 ≅ ∠3, by the vertical angles theorem.
(By the vertical angles theorem, vertical angles formed by two intersecting lines are congruent).
Then ∠2 ≅ ∠3, by the transitive property.
(Transitive property states that, if a = b and b = c, then a = c)
By the corresponding angles converse postulate, two lines p and q are parallel.
(If the corresponding angles of two lines cut by a transversal are equal, the lines are parallel).
Hence it is proved.
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If you roll the cube 30 times, how many 3's?
if you could show work that would be great ((:
The expected number of 3's would be 5.
What is probability?
Probability is a field of mathematics that calculates the likelihood of an experiment occurring. We can know everything from the chance of getting heads or tails in a coin to the possibility of inaccuracy in study by using probability.
The probability of rolling a 3 on a single roll of a fair six-sided die is 1/6, since there is one favorable outcome (rolling a 3) out of a total of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
However, it's important to keep in mind that the actual number of 3's obtained in 30 rolls of a die can vary, as the outcome of each roll is inherently random.
The expected value provides us with a guide for what to expect on average, but in any given sequence of rolls, the actual number of 3's may be higher or lower.
So, if you roll the cube 30 times, the expected number of 3's would be 30 * 1/6 = 5.
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Select all the equations where c=9c=9c, equals, 9 is a solution.
Choose 2 answers:
Choose 2 answers:
(Choice A)
A
4-c=54−c=54, minus, c, equals, 5
(Choice B)
B
20=14+c20=14+c20, equals, 14, plus, c
(Choice C)
C
15=c-615=c−615, equals, c, minus, 6
(Choice D)
D
\dfrac{c}{3}=3
3
c
=3start fraction, c, divided by, 3, end fraction, equals, 3
(Choice E)
E
36=4c36=4c
The equations that are solutions when c = 9 are B and C, and D and E.
When c = 9, the solutions for the equations are:
(A) 4 - c = 5: substituting 9 for c, we get 4 - 9 = -5, which is not true, so this equation is not a solution.
(B) 20 = 14 + c: substituting 9 for c, we get 20 = 14 + 9, which is true, so this equation is a solution.
(C) 15 = c - 6: substituting 9 for c, we get 15 = 9 - 6, which is true, so this equation is a solution.
(D) c/3 = 3: substituting 9 for c, we get 9/3 = 3, which is true, so this equation is a solution.
(E) 36 = 4c: substituting 9 for c, we get 36 = 4 * 9, which is true, so this equation is a solution.
So the equations that are solutions when c = 9 are B and C, and D and E.
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Answer:
Step-by-step explanation:
a father wants to buy toys for his daughter and his son. he wants to spend at least $25 on each child. if his overall budget to spend on both kids no more than $1 50, what is one possible combination of money the father can spend on each of his children for toys?
HELLPPP ILL MARK BRAINLIEST AND 40 POINTS
Answer:
Below
Step-by-step explanation:
Cross multiply the fractions to get
3 (x+4 ) = 4(x+3)
3x + 12 = 4x + 12 subtract 12 from both sides
3x = 4x <======obviously , this can only happen when x = 0
ONE SOLUTION
evaluate 4a - 2b +c over 2 abc when a=5 b=-2 and c=1
Answer:
0.85
Step-by-step explanation:
4a-2b+c/2abc
where a=5,b=2,c=1
4(5)-2(2)+1/2×5×2×1
20-4+1/20
17/20
0.86
If the sum of six consecutive odd integers is 504, what is the smallest of the six
integers?
Answer:
79
Step-by-step explanation:
You want the smallest of 6 consecutive odd integers that have a sum of 504.
AverageThe average of the integers will be the even integer between the middle two odd integers. The smallest odd integer is 5 less than this average.
The average is the sum divided by the number of integers:
504/6 = 84
The smallest of the 6 integers is 84 -5 = 79.
__
Additional comment
The integers are 79, 81, 83, 85, 87, 89.
Consecutive integer problems are often easily worked by considering the average of the integers, as here.
If you feel like you need an equation, you can let x represent the smallest. Then the sum is ...
x +(x+2) +(x+4) +(x+6) +(x+8) +(x+10) = 6x +30 = 504
6x = 474 ⇒ x = 474/6 = 79
Look at pic for directions and the triangle
9) What is the area of the triangle?
10) what is the area of the rectangular shaded (lavender) area that lies behind the triangle
11) explain how to find the area of the triangle without using a formula
For the given grid a. The area of the triangle is 21 square units. b. The area of the rectangle is 42 square units.
What is area?An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.
Given that the points on the grid are 1 units apart we have:
The height of the triangle is 6 units and the base of the triangle is 7 units.
The area of the triangle is:
A = 1/2 (b)(h)
A = 1/2(7)(6)
A = 21 square units.
The area of the triangle is 21 square units.
The length of the rectangular grid is 7 units and the width is 6 units.
The area of the rectangle is:
A = (l)(b)
A = (7)(6)
A = 42 square units.
The area of the rectangle is 42 square units.
Hence, a. The area of the triangle is 21 square units. b. The area of the rectangle is 42 square units.
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Consider the points A = (0, 0), B = (1, 0) and C = (0, 1). Now choose a random point (X, Y ) uniformly from the union of the line segments AB and AC. Find Cov(X, Y )
The answer is cov(x,y) = 0.059, if we choose a random point (X, Y ) uniformly from the union of the line segments AB and AC.
As given, A = (0, 0) ,B = (1, 0) and C = (0, 1)
and we have the select x and y from line segment AB and AC respectively.
So, x can be 1.0, 0.8, 0.6, 0.4, 0.2
and y can be 1.0, 0.8, 0.6, 0.4, 0.2
Now we have to find,
cov(x,y) = ∑(Xi - μ) (Yj - v) / n
where,
Cov(X, Y) represents the covariance of variables X and Y.Σ represents the sum of other parts of the formula.(Xi) represents all values of the X-variable.µ represents the average value of the X-variable.Yj represents all values of the Y-variable.v represents the average value of the Y-variable.Σ represents the sum of the values for both (Xi-µ) and (Yj-v).n represents the total number of data points across both variables.Now µ = 1.0 + 0.8 + 0.6 + 0.4 + 0.2 + 0.0 ÷ 6
µ = 3 ÷ 6
µ = 1/2 = 0.5
Now v = 1.0 + 0.8 + 0.6 + 0.4 + 0.2 + 0.0 ÷ 6
v = 3 ÷ 6
v = 1/2 = 0.5
Now, we have to find
(Xi - µ) = (1.0 - 0.5), (0. 8- 0.5), (0.6 - 0.5), (0.4 - 0.5), (0.2 - 0.5),
(0.0 - 0.5)
(Xi - µ) = 0.5, 0.3, 0.1, (-0.1), (-0.3), (-0.5)
(Yj - v) = (1.0 - 0.5), (0. 8- 0.5), (0.6 - 0.5), (0.4 - 0.5), (0.2 - 0.5),
(0.0 - 0.5)
(Yj - v) = 0.5, 0.3, 0.1, (-0.1), (-0.3), (-0.5)
Now calculate (Xi - μ) (Yj - v): multiply them with their respective one
(Xi - μ) (Yj - v) = (0.5×0.5), (0.3×0.3), (0.1×o.1), (-0.1)(-0.1), (-0.3)(-0.3),
(-0.5)(0.5)
therefore,
∑(Xi - μ) (Yj - v) = (0.25+ 0.09 + 0.01 + 0.01 + 0.09 + 0.25)
∑(Xi - μ) (Yj - v) =0.35
Now, cov(x,y) = ∑(Xi - μ) (Yj - v) / n
cov(x,y) = 0.35 / 6
cov(x,y) = 0.059
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33) a ten year old is having his birthday party at the roller rink. Explain two different counts that his parents will estimate for the party. One should be an over-estimate and the other can be an under-estimate.
34) an airline pilot is planning a fight from new York’s LaGuardia airport to Boston, mass. Explain two measurements the pilot must not estimate in order to plan a safe flight
35) a construction work needs to build a concrete wall to support a prefabricated ramp that will be used as a handicap access to the public library. (A) explain why this would not be a place to estimate measurements. (B) describe where some estimates might be adequate for the plan
Measurements are important for a safe and secure structure, as they must adhere to local building codes and regulations. Preliminary planning can be used to estimate the height, width, and length of the ramp and the amount of concrete and other materials needed.
What is a measurement?A measurement is a measurable quantity that includes length, breadth, height, area, volume, weight, or time. Measurements are used to measure an object's or area's size, shape, or qualities.
A) Estimating measurements for a construction project such as this is not advisable, as accuracy is essential for ensuring the wall is stable and safe for use. Calculations must be precise in order to ensure the dimensions, material strength, and angles are correct, which is why it is important to use exact measurements in the plans.
B) In some cases, estimates may be acceptable. For example, the approximate number of blocks or pieces of material needed may be estimated based on the overall size and shape of the wall. Additionally, estimates of the time needed to complete the project may be made based on the experience of the construction team and the complexity of the project.
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There are 3 notebooks for $3.29. How much is one notebook?
Answer:$1.097
Step-by-step explanation:
3.29/3
then you will get 1.0967 (round up)
2/9 • 3/1
Fraction and whole number multiplication! Image attached
Soledad buys 5 ounces of frozen yogurt for $2.25. What is the unit price of the frozen yogurt in dollars per ounce?
Answer:
0.45
Step-by-step explanation:
You divided 2.25 by 5
A triangular prism has a height of 5.9 meters and volume 86.376 cubic meters. What is the area of the base of the prism?
Answer: 14.64m^2
Step-by-step explanation:
Base area = volume/height
Substitute:
86.376/5.9 = 14.64cm^2
Hope that helps!
11. Mr. Lee made finding the zeros of the polynomial function h(x) = x(x-4) (x + 2). He stated that
the zeros of this polynomial are x = 1, x = 4, x = -2. Explain his error. (DOK3)
Answer:
one of the zeros is x = 0 and not x = 1
Step-by-step explanation:
Zeros are found by setting each of the factors to 0 and solving for x
Factors are x, x -4, x + 2
Set each of them to 0:
x =0
x + 2 = 0 ==> x = -2
x -4 = 0 ==> x = 4
So one of the zeros is x = 0 and not x = 1
800 people attended a football game. If 20% of the people who attended were teenagers, how many teenagers attended the game? Insert the values given in the problem then scale up or down to find the missing value. attendees percent 100 attempt 3 out of 3 Emilia Salkic Proportional Reasoning, Percents (Level 2) Feb 03, 2:20:32 PM 800 people attended a football game. If 20% of the people who attended were teenagers, how many teenagers attended the game?
Using percentages, we can deduce that 160 of the 800 attendees at the football and math conference were teenagers or 20% of the total.
What are percentages?In mathematics, a value or ratio that can be expressed as a fraction of 100 is known as a percentage.
When determining a percentage of a number, we should first divide it by 100 before multiplying the result by 100.
Therefore, the proportion refers to a component per 100.
The word percent denotes a percentage of 100.
A percentage (%) is a number that represents how many out of 100 it is and can also be written as a decimal or fraction.
To convert a percentage to a fraction, place the percentage's value in the numerator and 100 in the denominator.
So, we are aware of:
The football game attracted 800 spectators.
20% of the population were teenagers.
The number of teens at that time was:
= 800/100*20
= 8*20
= 160
Therefore, using percentages, we can deduce that 160 of the 800 attendees at the football and math conference were teenagers, or 20% of the total.
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PLEASE HELP ME PLEASE THANK YOU
PART A
Craig went bowling with $35.50 to spend. He rented the shoes for $6.50 and paid $7.25 for each game.WRITE AN EQUATION TO DETERMINE X,THE GREATEST NUMBER OF GAME CRAIG COULD HAVE PLAYED. ONLY WRITE EQUATION HERE
Equation:__________
PART B
Determine what the greatest number of games he can play is. SHOW YOUR WORK
Answer:_________Games
Answer:
Step-by-step explanation:
4
for the stem and leaf plot created in question 9 are there any extreme values? if so, what are they?
As a result, the extreme stem values may contain fewer rows than the other stem values. In our graph, the extreme stem values are 1 and 4, and they both have fewer rows than the middle values (2 and 3).
The shape and spread of a continuous data distribution are depicted by stem and leaf plots. These graphs are similar to histograms, but instead of bars, they display numbers.
Statistical software packages use an algorithm to improve our ability to read stem and leaf plots by displaying multiple rows of each stem value based on the properties of the data, as shown in the body fat percentage graph. There are two ones, five twos, five threes, and four fours.
The graph divides stem values into five rows for the body fat percentage data. Each row only has two leaf values (e.g., 0 and 1, 2 and 3, etc.) The leaf values terminate at the dataset's minimum and maximum values.
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Please please help
7 A cylinder of height h metres is inscribed in a sphere of constant radius R metres.
a If the cylinder has radius r metres, show that r? = R? - ]h?.
b Show that the volume of the cylinder is given by V = Ih (4R? - h).
c Show that the volume of the cylinder is maximised when h = §v3R.
d Hence show that the ratio of the volume of the sphere to the maximum volume of the cylinder is V3: 1.
12 A canned fruit producer wishes to minimise the area of sheet metal used in manufacturing cylindrical cans of a given volume. Find the ratio of radius to height for the desired can.
10 A rectangle is inscribed in a quadrant of a circle of radius r so that two of its sides are along the bounding radii of the quadrant.
a If the rectangle has length x and width y, show that its area is given by A = yVr2 - y
b Show that the maximum possible area of the rectangle is †r?
Answer:
7a) To show that r^2 = R^2 - h^2, one can use the Pythagorean theorem and the fact that the distance from the center of the sphere to the surface of the cylinder is h/2.
7b) To find the volume of the cylinder, one can use the formula for the volume of a cylinder, V = πr^2h, where r is the radius and h is the height.
7c) To find the maximum volume of the cylinder, one can differentiate the expression for the volume with respect to h, set the derivative equal to zero, and solve for h.
7d) The ratio of the volume of the sphere to the maximum volume of the cylinder can be found by dividing the volume of the sphere by the maximum volume of the cylinder.
To find the ratio of radius to height for the desired can, one can set up an expression for the surface area of the can and differentiate it with respect to the radius and height. Setting the partial derivatives equal to zero, one can find the critical points, and evaluate which one is the minimum.
10a) To find the area of the rectangle, one can use the formula for the area of a rectangle, A = xy, where x is the length and y is the width.
10b) To find the maximum area of the rectangle, one can differentiate the expression for the area with respect to x and y, set the partial derivatives equal to zero, and solve for x and y.
Step-by-step explanation: