Answer:
The 99% confidence interval for the proportion of all students at this school who have their names written on their graphing calculators is (0.4652, 0.8148).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
A random sample of 50 students is selected, and of the students questioned, 32 had their names written on their graphing calculators.
This means that [tex]n = 50, \pi = \frac{32}{50} = 0.64[/tex]
99% confidence level
So [tex]\alpha = 0.01[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.64 - 2.575\sqrt{\frac{0.64*0.36}{50}} = 0.4652[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.64 + 2.575\sqrt{\frac{0.64*0.36}{50}} = 0.8148[/tex]
The 99% confidence interval for the proportion of all students at this school who have their names written on their graphing calculators is (0.4652, 0.8148).
A focus group study has always been the primary way for ABC Toys Company to gather customers' opinions for marketing strategy. In the past, when a new product was popular in the market, a focus group study had incorrectly indicated that it would be unpopular with a probability of 0.275. When a new product was unpopular, a focus group had incorrectly indicated that it would be popular with a probability of 0.175. Assume that the probability of a new product being popular in the market is 0.60. What is the probability that a new product will be unpopular if a focus group study indicates that it will be unpopular?
Using conditional probability, there is a 0.6667 = 66.67% probability that a new product will be unpopular if a focus group study indicates that it will be unpopular.
What is Conditional Probability?Conditional probability is the probability of one event happening, considering a previous event. The formula is:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which:
P(B|A) is the probability of event B happening, given that A happened.[tex]P(A \cap B)[/tex] is the probability of both A and B happening.P(A) is the probability of A happening.For this problem, the events are given as follows:
Event A: Study indicates that the product is unpopular.Event B: Product is unpopular.The parameters are given as follows:
P(A) = 0.6 x 0.275 + 0.4 x 0.825 = 0.495.[tex]P(A \cap B) = 0.4 \times 0.825 = 0.33[/tex]Hence the conditional probability is:
[tex]P(B|A) = \frac{0.33}{0.495} = 0.6667[/tex]
0.6667 = 66.67% probability that a new product will be unpopular if a focus group study indicates that it will be unpopular.
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graph
Y=-2x-3 and y=-x+6
Answer:
here ya go :)
the black line is y=-2x-3
the lavender line is y=-x+6
Given b(x)=|x+41|, what is b(-10)?
Answer:
31
Step-by-step explanation:
Substitute -10 for x in b(x)=|x+41|, obtaining f(-10) = |-10 + 41|. This result simplifies to f(-10) = |31| = 31
need help with this
Answer:
[tex]x^4-20x^3 +175x^2 -750x+1250[/tex]
Step-by-step explanation:
By the conjugate root theorem, 5+5i is also a root.
[tex]f(x)=a(x-5)^2 (x-5-5i)(x-5+5i) \\ \\
= a(x^2 - 10x+25)((x-5)^2 - (5i)^2) \\ \\
= a(x^2 - 10x+25)(x^2 - 10x+25+25) \\ \\
= a(x^2-10x+25)(x^2-10x+50) \\ \\
= a(x^4- 10x^3 + 50x^2 - 10x^3 + 100x^2 - 500x + 25x^2 - 250x+1250) \\ \\
= a(x^4-20x^3 +175x^2 -750x+1250) \\ \\ [/tex]
Which graph represents the solutions to the inequality |2x − 6| < 4? (5 points) number line with a closed circle on 1, shading to the left and a closed circle on 5, shading to the right number line with a closed circle on 1, shading to the right and a closed circle on 5, shading to the left number line with an open circle on 1, shading to the left and an open circle on 5, shading to the right number line with an open circle on 1, shading to the right and an open circle on 5, shading to the left
The graph that represents the inequality is:
Number line with an open circle on 1, shading to the right and an open circle on 5, shading to the left.
What is the inequality?The inequality is:
|2x - 6| < 4
Which means that the distance of 2x - 6 to the origin is less than 4, hence:
-4 < 2x - 6 < 4.
The solution is:
2x - 6 > -4
2x > 2
x > 1
2x - 6 < 4
2x < 10
x < 5
The solution has open circles at x = 1 and x = 5, as they are not part of the solution, and the shading is in the middle, as the solution is 1 < x < 5, hence the correct option is:
Number line with an open circle on 1, shading to the right and an open circle on 5, shading to the left.
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Suppose we want to choose 5 letters, without replacement, from 15 distinct letters.
[tex]order \: does \: not \:matter \\ sample \: space = 15 \: letters \\ no \: repetition\\ P(A) = 15C5 = 3003 \: ways[/tex]
What is the answer to this system of equations? 2x - 3y= 21 and -6x + 2y = 7.
answer:
x = - 9/2
y = -10
Step-by-step explanation:
make both equations look like y =
2x-3y=21 -> -3y= -2x+21 -> y = (-2x+21)/-3 -> y = 2/3x - 7
-6x + 2y = 7 -> 2y = 6x + 7 -> y = 3x + 7/2
now make the equations equal to each other (since they both equal y)
get x
2/3x - 7 = 3x + 7/2
-7 -7/2 = 3x - 2/3x
-14/2 -7/2 = 9/3x - 2/3x
-21/2 = 7/3x
(3/7) -21/2 = x
x = (3/7) -21/2
x = - 63/14
x = - 9/2
put it back in an equation to get y
y = 3x + 7/2
y = 3(-9/2) + 7/2
y = -27/2 + 7/2
y = -20/2
y = - 10
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100 POINTS
Find the volume of the composite solid.
A ) 1099.01 m^3
B) 104.67 m^3
C) 889.67 m^3
D) 785 m^3
Volume of a cylinder: PI x radius^2 x height
= 3.14 x 5^2 x 10 = 785 m^3
Volume of a cone: pi x radius^2 x height/3
= 3.14 x 5^2 x 4/3 = 104.67 m^3
Total volume = 785 + 104.67 = 889.67 M63
Answer: C. 889.67 M^3
Answer:
[tex]\textsf{C)}\quad \sf 889.67\: m^3[/tex]
Step-by-step explanation:
The given composite solid is made up of a cylinder and a cone.
To find the volume of the composite solid, find the volume of the cylinder and the volume of the cone, then add them together.
Volume of a cylinder:
[tex]\sf V=\pi r^2 h[/tex]
(where r is the radius and h is the height)
Given:
r = 5 mh = 10 mSubstitute the given values into the formula and solve for V:
[tex]\begin{aligned}\implies \sf Volume\:of\:the\:cylinder & = \sf \pi (5)^2(10)\\ & = \sf 250 \pi \:\:m^3 \end{aligned}[/tex]
Volume of a cone:
[tex]\sf V=\dfrac{1}{3} \pi r^2 h[/tex]
(where r is the radius and h is the height)
Given:
r = 5 mh = 4 mSubstitute the given values into the formula and solve for V:
[tex]\begin{aligned}\implies \sf Volume\:of\:the\:cone & = \sf \dfrac{1}{3} \pi (5)^2(4)\\ & = \sf \dfrac{100}{3}\pi \:\:m^3 \end{aligned}[/tex]
Therefore, the volume of the composite solid is:
[tex]\begin{aligned}\implies \sf Volume\:of\:composite\:solid & = \sf Cylinder\:volume+Cone\:volume\\ & = \sf 250\pi + \dfrac{100}{3}\pi \\ & = \sf \dfrac{850}{3} \pi \\ & = \sf \dfrac{850}{3} \times 3.14 \\ & = \sf 889.67\:m^3\:(2\:d.p.)\end{aligned}[/tex]
I have the question shown in the screenshot
The width and height of the rectangle inscribed in the right triangle have a measure of 3.529 units.
How to find the dimensions of the rectangle of maximum area by optimization
In this problem we must use critical values and algebraic methods to determine to determine the dimensions of the rectangle such that the area is a maximum. The equation of the quadrilateral is formed by definition of the area of a rectangle:
A = w · h (1)
Where:
w - Width of the rectangle.h - Height of the rectangle.And the area of the entire triangle is:
0.5 · (5) · (12) = w · h + 0.5 · w · (12 - h) + 0.5 · (5 - w) · h
30 = w · h + 6 · w - 0.5 · w · h + 2.5 · h - 0.5 · w · h
30 = 6 · w + 2.5 · h
2.5 · h = 30 - 6 · w
h = 12 - 2.4 · w (2)
The quadrilateral of maximum area is always a square, then we must solve for w = h:
w = 12 - 2.4 · w
3.4 · w = 12
w = 3.529
Then, the width and height of the rectangle inscribed in the right triangle have a measure of 3.529 units.
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f(x)=√x+11. Find the inverse of f(x).
Answer:
f(x)⁻¹ = (x - 11)²
Step-by-step explanation:
To find the inverse of the function, you need to (1) swap the places of the "x" and "y" variables and then (2) solve for "y". Remember, f(x) is another way of writing "y".
y = √x + 11 <----- Original equation
x = √y + 11 <----- Swap variables
x - 11 = √y <----- Subtract 11 from both sides
(x - 11)² = (√y)² <----- Square both sides
(x - 11)² = y <----- Simplify
Another way of writing the output of an inverse function is with f(x)⁻¹.
PLSSSSSSSSSSSSSSS HELP pls
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Let's solve ~
[tex]\qquad \sf \dashrightarrow \: 10x + 30 = 90[/tex]
[ according to given figure ]
[tex]\qquad \sf \dashrightarrow \: 10x = 90 - 30[/tex]
[tex]\qquad \sf \dashrightarrow \: 10x = 60[/tex]
[tex]\qquad \sf \dashrightarrow \: x = 60 \div 10[/tex]
[tex]\qquad \sf \dashrightarrow \: x = 6 \degree[/tex]
Correct choice is D
I have no clue what this question is asking nor how to answer it
Which of the following numbers are solutions of the sentence x-3< 2?
| -3
|| 0
||| 2
IV 5
II only
III only
I and III only
I, II, and III only
Or
I, II, III, and IV
Answer:
IV 5
Step-by-step explanation:
you have to solve for x:
x-3<2
x-3+3<2+3
x=5
could you put the brainlyest Icon for me? I have 1500 points, but no brainlyest, so I can't move up the ranks.
Paul's budget for food is 30% of the budget he has for rent.
If his budget for food is $225, what is his budget for rent?
Paul's rent budget is $750, given that his food budget of $225, is 30 percent of his rent budget.
What are fractions?Fractions are representation of quantities as a part of a certain quantity.
Written as a/b, read as a by b, and functions as a divided by b, represents a fraction showing a parts of b equal parts.
a is called the numerator, and b is called the denominator.
What are percentages?Percentages are modified fractions where the whole quantity is represented by 100.
To convert a fraction into percentages we multiply it by 100, while to convert percentages to fraction divided by 100.
How to solve the question?In the question, we are given that Paul's budget for food is 30% of the budget he has for rent.
We are asked if his budget for food is $225, what is his budget for rent.
We assume Paul's budget for rent to be $x.
We know that Paul's food's budget is 30 percent of his rent.
Hence, Paul's food budget = 30% of x.
or, Paul's budget for food = 30/100 * x {To convert percentages to fraction divided by 100}
But, we know that Paul's food budget is $225. Thus, we can show that:
225 = 30/100 * x,
or, x = (225*100)/30 = 750.
Thus, Paul's rent budget is $750, given that his food budget of $225, is 30 percent of his rent budget.
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.
A new refrigerator that normally sells for $897.00 is on sale for 15% off. If the sales tax is 7.5%, then how much will you pay in total for the refrigerator while it’s on sale?
Answer: About 830 dollars
Step-by-step explanation: find 15% off on 897 then find 7.5 percent of 897, then add them together.
I
23
12
N
L
12
23
O
K
M
18
18
Answer: [tex]\sqrt{407}[/tex]
Step-by-step explanation:
In this triangle, we know that [tex]IJ=46, IK=24, JK=36[/tex].
So, using the Law of Cosines in triangle IJK,
[tex]36^2 = 46^2 + 24^2 - 2(46)(24) \cos (\angle JIK)\\\\-1396=-2(46)(24) \cos(\angle JIK)\\\\\cos (\angle JIK)=\frac{349}{552}[/tex]
Using the Law of Cosines in the triangle LIK,
[tex](KL)^2 = 23^2 + 24^2 - 2(23)(24) \cos (\angle JIK)\\\\(KL)^2 = 23^2 + 24^2 - 2(23)(24)\left(\frac{349}{552} \right)\\\\(KL)^2 = 407\\\\KL=\sqrt{407}[/tex]
g(r) = -20 +11r
8(11) = ?
Answer:
g(11) = 101
Step-by-step explanation:
g(r) = -20 + 11r
So in g(11), we just need to replace 'r' with '11'
g(11) = -20 + 11 (11)
--> g(11) = -20 + 121 = 101
--> g(11) = 101
Please comment down below if 8(11) wasn't a typo. I thought it was, so I replaced 8 with g. If that was intentional, I'll edit my answer as soon as possible.
Mark and paula are storing masks. Paula has 70% of the total number of masks. If paula has 840 more masks thatn mark, how many masks should paula give to mark si they both have an equal number of masks
Answer:
1050 masks per person.
Step-by-step explanation:
1) Create an equation to find the total number of masks
2) Solve equation and divide total by 2 to get the number of masks that can be given to both Mark and Paula so that they have equal number.
1) Since Paula has 70% of the total masks, it can be represented as 0.7 * t where t is the variable representing the total number of masks. Mark on the other hand has the remaining number of masks out of 100% which is 30%. We can express that as 0.3. We multiply that by t because he has 30% of the masks. The final equation we have is 0.3t + 840 = 0.7t. 840 because Paula has 840 more masks than Mark.
2) Solve by subtracting 0.3t from both sides leaving you with 0.4t = 840. We divide both sides by 0.4 to get 2100. That is the total number of masks. If we divide that number by 2, we get 1050.
See attachment for the full question
The inverse of the demand function is; P = 9 - 0.25Q
The profit-maximizing price and quantity are; $8.5 and 2 units.
The maximum profit is; $1
How to find the inverse of a function?
A) The demand function we are given is;
Q = 36 - 4P
Making P the subject gives the inverse demand function;
P = (36 - Q)/4
P = 9 - Q/4
P = 9 - 0.25Q
B) The profit-maximization point is the point at which MR = MC.
MR refers to the marginal revenue and MC is the marginal cost.
MC can be calculated as the first derivative of the cost function:
C(Q) = 4 + 4Q + Q²
MC = C'(Q) = 2Q + 4
Total Revenue = Price * Quantity
Total Revenue = (9 - 0.25Q) * Q
Total Revenue = 9Q - 0.25Q²
MR is gotten by differentiating Total Revenue to get;
MR = 9 - 0.5Q
Applying the condition MR = MC, we have;
9 - 0.5Q = 4 - 2Q
Solving for Q gives Q = 2
Thus, profit maximizing quantity is 2.
Thus, profit maximizing price will be;
P(2) = 9 - 0.25(2)
P(2) = $8.5
C) Formula for Maximum Profit is;
Profit = Total Revenue - Total Cost
Total Revenue = 8.5 * 2
Total revenue = $17
Total Cost is;
C(2) = 4 + 4(2) + 2²
C(2) = $16
Thus;
Maximum Profit = 17 - 16 = $1
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The average heart rate of an adult human is 72 beats per minute. For an adult elephant, the average heart rate is 35 beats per minute.
A. Whose heart beats more times in one hour
B. Whose heart makes 1,000,000 beats in less time
Calculate a and b if
[tex] \sqrt{ \frac{ {7}^{2014} - {7}^{2012} }{12} } = a( {7}^{b} )[/tex]
and a is not a multiple of 7.
Notice that
[tex]7^{2014} - 7^{2012} = 7^{2012} \bigg(7^2 - 1\bigg) = 7^{2012} \times 48 = 2^4 \times 3 \times 7^{2012}[/tex]
[tex]12=2^2\times3[/tex], so
[tex]\dfrac{7^{2014}-7^{2012}}{12} = 2^2 \times 7^{2012}[/tex]
Then taking the positive square root gives
[tex]\sqrt{\dfrac{7^{2014}-7^{2012}}{12}} = \sqrt{2^2 \times 7^{2012}} = 2\times7^{1006}[/tex]
so [tex]\boxed{a=2}[/tex] and [tex]\boxed{b=1006}[/tex].
Which equation is represented by the graph below?
A y=In x-3
B y=In x-4
C y=e^3 -3
D y=e^3 -4
Answer:
D
Step-by-step explanation:
I assume that for C and D, you meant e^x.
Since the function is defined at 0, eliminate A and B (since ln 0 is undefined)
Between C and D, we know that when x=0, e^x = 1. Since the y intercept of the graph is -3, this means the equation is y = e^x - 4.
Aaron wants to make All-Tournament Team for his high school
basketball team. He needs to average at least 24 points a game
to get selected for All-Tournament Team. They are playing 4
games, in the first 3 games he scored 30, 17, and 25 points.
Answer:24
Step-by-step explanation:
Answer:
Step-by-step explanation:
What is the difference between 89.4 and 13.621 do not round your answer
Answer:
89.4 - 13.621 = 75.779
Explanation:
Subtract 13.621 from 89.4 to get 75.779.
FInd the solution for the ordered pair
2x + y = 5 and x + y = 6
Answer:
(-1,7)
Step-by-step explanation:
Use elimination, x = -1, y = 7
Answer:
(-1, 7)
Step-by-step explanation:
Elimination Method:Elimination: Step 1
Write in standard form (you can have negatives)
(standard form= Ax+By=C)
Elimination: Step 2
Create(find) opposites
Elimination: Step 3
Add to cancel one variable
Elimination: Step 4
Solve for variable
Elimination: Step 5
Substitute into other equation to find other variable
Substitution (Method)
In a linear system of equations, substitution results in one equation with one variable. The solution to a linear system is an ordered pair, not just a single value.
Problem:
Solve 2x+y=5;x+y=6
Steps:
I will solve your system by substitution.
(You can also solve this system by elimination.)
2x+y=5;x+y=6
Step: Solve 2x+y=5 for y:
2x+y+−2x=5+−2x(Add -2x to both sides)
y=−2x+5
Step: Substitute −2x+5 for yin x+y=6:
x+y=6
x+−2x+5=6
−x+5=6(Simplify both sides of the equation)
−x+5+−5=6+−5(Add -5 to both sides)
−x=1
-x/-1 = 1/-1 (Divide both sides by -1)
x=−1
Step: Substitute −1 for x in y=−2x+5:
y=−2x+5
y=(−2)(−1)+5
y=7(Simplify both sides of the equation)
Answer:
x=−1 and y=7
The circumference of a circle is 13π in. What is the area, in square inches? Express your answer in terms of \piπ.
The area of the circle with the given circumference is: 42.25π in.².
What is the Area and Circumference of a Circle?Area = πr²
Circumference = 2πr.
Given the following:
Circumference of the circle = 13π in.
Find the radius (r):
2πr = 13π
Divide both sides by 2π
2πr/2π = 13π/2π
r = 13/2
r = 6.5
Area = πr² = π(6.5²) = 42.25π in.²
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Select the solution to the following system of equations:
4x + y = 6
2x - 3y = -4
The solution to the given system of linear equations is x = 1, y = 2. That is (1, 2)
Simultaneous linear equationsFrom the question, we are to solve the given system of equations
The given system of equations is
4x + y = 6 -----------(1)
2x - 3y = -4 -----------(2)
From the equation (1)
4x + y = 6
y = 6 - 4x ----------(3)
Substitute the value of y into equation (2)
2x - 3y = -4
2x -3(6 - 4x) = -4
2x -18 +12x = -4
2x + 12x = -4 + 18
14x = 14
x = 14/14
x = 1
Substitute the value of x into equation (3)
y = 6 - 4x
y = 6 - 4(1)
y = 6 - 4
y = 2
Hence, the solution to the given system of linear equations is x = 1, y = 2. That is (1, 2)
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What is the product?
The correct product of (6x - 2)(6 x + 2) is 36x^2 - 4
How to determine the product?The expression is given as:
(6x - 2)(6 x + 2).
The above expression is a difference of two squares.
And this is represented as
(a - b)(a + b)= a^2 - b^2
So, we have
(6x - 2)(6 x + 2) = (6x)^2 - 2^2
Evaluate
(6x - 2)(6 x + 2) = 36x^2 - 4
Hence, the correct product of (6x - 2)(6 x + 2) is 36x^2 - 4
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Complete question
What is the product?
(6x - 2)(6 x + 2).
Maritza desea ir de viaje con su familia, que está conformada por ella, su esposo y sus 3 hijos. Ella comprará pasajes cuyo costo es de S/75 por persona (sólo ida), también desea comprar un paquete turístico para cuatro personas cuyo valor es de S/800, además contrató 3 habitaciones por el precio de S/60 cada una. También se proyectó gastar en alimentación 150 diarios y una bolsa de viaje de S/130 por persona. Si su viaje es de 5 días y 4 noches y actualmente ella cuenta con S/1000 y su esposo con S/1800, ¿Cuánto dinero les falta para cubrir todos los gastos de su viaje?
El dinero que le falta a Maritza y su esposo para viajar es 330.
¿Cuánto dinero les falta?Para calcular el dinero que les falta debemos identificar la cifra total que les va a costar el viaje:
75 x 10 = 75080060 x 3 = 180 150 x 5 = 750130 x 5 = 650750 + 800 + 180 + 750 + 650 = 3,130El valor total del viaje es s/3,130, entonces les faltarían s/330.
3,130 - 2,800 = 330
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Find the area of the region that lies inside both curves.
r² = 2 sin(2θ), r=1
The area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
How to find the area of the region that lies inside both curves?Since the curves are
r² = 2sin(2θ) and r = 1.We find their point of intersection.
So, r² = r²
2sin(2θ) = 1²
2sin(2θ) = 1
sin(2θ) = 1/2
2θ = sin⁻¹(1/2)
2θ = π/6
θ = π/12
So, we integrate the area from θ = 0 to θ = π/12
Now the area A of the region between two curves between θ = α to θ = β is
[tex]A = \int\limits^{\beta }_\alpha ({r^{2} - r^{'2} }) \, d\theta[/tex]
So, the area betwwen the curves r² = 2sin(2θ), r = 1 between θ = 0 to θ = π/12 is
[tex]A = \int\limits^{\frac{\pi}{12} }_0 ({r^{2} - r^{'2} }) \, d\theta \\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1^{2} }) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1}) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 {2sin2\theta}\, d\theta - \int\limits^{\frac{\pi}{12} }_0 {1} \, d\theta\\= [2(-cos2\theta)/2 - \theta]_{0}^{\frac{\pi}{12} } \\= [-cos2\theta - \theta]_{0}^{\frac{\pi}{12} } \\= -cos2\frac{\pi}{12} - \frac{\pi}{12} - ( -cos2(0) - 0)\\[/tex]
[tex]= -cos2(\frac{\pi}{12}) - \frac{\pi}{12} - ( -cos2(0) - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -cos0 - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -1)\\= -(\frac{\sqrt{3} }{2} ) - \frac{\pi}{12} + 1\\= -0.8660 - 0.2618 + 1\\= -1.1278 + 1\\= -0.1278[/tex]
So, the area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
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Compute without a calculator:
$$(72\cdot 78\cdot 85\cdot 90\cdot 98)\div (68\cdot 84\cdot 91\cdot 108).$$
(There's an easier way than multiplying out the giant products $72\cdot 78\cdot 85\cdot 90\cdot 98$ and $68\cdot 84\cdot 91\cdot 108$!)
Take the prime factorization of each factor in the product.
[tex]\dfrac{72 \times 78 \times 85 \times 90 \times98}{68 \times 84 \times 91 \times 108} \\\\ ~~~~~~~~ = \dfrac{(2^3\times3^2)\times(2\times3\times13)\times(5\times17)\times(2\times3^2\times5)\times(2\times7^2)}{(2^2\times17)\times(2^2\times3\times7)\times(7\times13)\times(2^2\times3^3)} \\\\ ~~~~~~~~ = \dfrac{2^6 \times 3^5 \times 5^2 \times 7^2 \times 13 \times 17}{2^6 \times 3^4 \times 7^2 \times 13 \times 17} \\\\ ~~~~~~~~ = 3 \times 5^2 = \boxed{75}[/tex]