There are 20 students on the school's student council. A special homecoming dance committee is to be formed by randomly selecting 7 students from student council. How many possible committees can be formed?
Final answer:
Using the combinations formula, there can be 77,520 different committees formed by selecting 7 students from a student council of 20 students.
Explanation:
To determine the number of possible committees that can be formed by selecting 7 students from a group of 20 students, we will use the combinations formula since the order of selection does not matter. This is a classic example of a combinatorial problem where we are choosing a subgroup from a larger group without regard to the order in which they are chosen.
The formula for combinations is as follows:
C(n, k) = n! / (k! * (n - k)!)
Where:
n is the total number of items,
k is the number of items to choose,
! indicates factorial, which means the product of all positive integers up to that number. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Applying this formula to our problem:
C(20, 7) = 20! / (7! * (20 - 7)!) = 20! / (7! * 13!) = (20 x 19 x 18 x 17 x 16 x 15 x 14) / (7 x 6 x 5 x 4 x 3 x 2 x 1)
After simplifying the factorial expressions and canceling out common factors, we find the number of possible committees that can be formed.
Therefore, there are 77520 possible committees that can be formed from a student council of 20 students by selecting 7.
Marla swims twice a week. her equipment cost her $32.85 and she has a membership to the pool for $35 plus #3.50 per visit. she can walk to the pool, so transportation is free. how much will swimming cost her for the year?
Solve the triangle. B=36 a=41 c=20
In the triangle with B = 36°, a = 41, c = 20, b ≈ 45.3, C ≈ 29.5, A ≈ 114.5.
To solve the triangle, we can use the Law of Sines and the fact that the sum of angles in a triangle is 180 degrees.
Given:
[tex]- Angle \( B = 36° \)[/tex]
[tex]- Side \( a = 41 \)[/tex]
[tex]- Side \( c = 20 \)[/tex]
First, we need to find angle [tex]\( A \)[/tex]. We can use the fact that the sum of angles in a triangle is 180 degrees:
[tex]\[ A = 180 - B - C \][/tex]
We can find angle [tex]\( C \)[/tex] using the Law of Sines:
[tex]\[ \frac{\sin C}{c} = \frac{\sin B}{b} \][/tex]
Solving for [tex]\( C \):[/tex]
[tex]\[ \sin C = \frac{c \times \sin B}{b} \][/tex]
[tex]\[ \sin C = \frac{20 \times \sin 36°}{b} \][/tex]
[tex]\[ \sin C = \frac{20 \times 0.5878}{b} \][/tex]
[tex]\[ \sin C = \frac{11.756}{b} \][/tex]
Now we can find angle [tex]\( C \)[/tex] using the inverse sine function:
[tex]\[ C = \sin^{-1}\left(\frac{11.756}{b}\right) \][/tex]
Now, we can substitute [tex]\( C \) i[/tex]nto the equation for [tex]\( A \)[/tex] and solve for [tex]\( b \):[/tex]
[tex]\[ A = 180 - B - C \][/tex]
[tex]\[ A = 180 - 36 - \sin^{-1}\left(\frac{11.756}{b}\right) \][/tex]
[tex]\[ A = 144 - \sin^{-1}\left(\frac{11.756}{b}\right) \][/tex]
Since we know all three angles of the triangle, we can find [tex]\( b \)[/tex] using the Law of Sines again:
[tex]\[ \frac{\sin A}{a} = \frac{\sin B}{b} \][/tex]
[tex]\[ \sin A = \frac{a \times \sin B}{b} \][/tex]
[tex]\[ \sin A = \frac{41 \times \sin 36}{b} \][/tex]
[tex]\[ \sin A = \frac{41 \times 0.5878}{b} \][/tex]
[tex]\[ \sin A = \frac{24.0678}{b} \][/tex]
Now, we can find [tex]\( b \)[/tex] using the inverse sine function:
[tex]\[ b = \frac{24.0678}{\sin A} \][/tex]
Finally, we'll use the given values and calculate the closest option:
[tex]\[ A \approx 144 - \sin^{-1}\left(\frac{11.756}{b}\right) \][/tex]
[tex]\[ b \approx \frac{24.0678}{\sin A} \][/tex]
[tex]\[ c \approx 20 \][/tex]
B = 36°
[tex]\[ a = 41 \][/tex]
After the calculation, the closest option is:
[tex]\[ b \approx 45.3, \ C \approx 29.5, \ A \approx 114.5 \][/tex]
Complete Question:
Solve the triangle.
B = 36°, a = 41, c = 20
choices:
b ≈ 27.5, C ≈ 115.5, A ≈ 28.5
b ≈ 45.3, C ≈ 25.5, A ≈ 118.5
b ≈ 45.3, C ≈ 29.5, A ≈ 114.5
b ≈ 27.5, C ≈ 25.5, A ≈ 118.5
the answer to the triangle the hexagon and the Diamond
When combining functions, what operation requires that you restrict the domain of one or both functions?
Pick the geometric term that matches the real-word object: a lip-gloss tube.
Answer: Cylinder.
Step-by-step explanation: Lip-gloss tubes may have different shapes, but the usual form of a lip-gloss tube is, as the name says, a tube.
This means that it has a circular base and a circular top, and a given height that connects the base and the top, the geometrical shape that has this shape is called a cylinder.
What is the distance between -14 and 5
Alexandra Romar has a previous balance at Porter Pharmacy of $68.42. She had payments and credits of $18.25. The monthly finance charge is 1.85% of the unpaid balance. After the finance charge was calculated, she made $34.00 in new purchases. What is her new balance?
Answer:
Her new balance is $85.10.
Step-by-step explanation:
Alexandra Romar has a previous balance at Porter Pharmacy = $68.42
She had payments and credits = $18.25
Now the unpaid balance = 68.42 - 18.25 = $50.17
The monthly finance charge on unpaid balance = 1.85% × 50.17
= [tex]\frac{1.85}{100}[/tex] × 50.17
= 0.928145 ≈ $0.93
So the balance with finance charge = 50.17 + 0.93 = 51.10
She made a new purchase = $34.00
Her new balance = 51.10 + 34.00 = $85.10
Her new balance is $85.10.
Three subtracted from x is less than or equal to -17
Please solve. best answer gets rewarded! show step by step.
Find the missing term. If y = 1 + i, then y3 − 3y2 + (-3y) (-3y + 1) (3y) (3i ) − 1 = -i .
Answer:3y is the answer
Step-by-step explanation:
what is the value of y?
Answer:
The correct option is D) 60°
Step-by-step explanation:
Consider the provided triangle.
From the provided figure it is given that one angle is 60° and we need to find the value of y.
As we know the sum of all interior angle in a triangle is 180°.
y°+y°+60°=180°
Now solve the above equation to find the value of y.
2y°+60°=180°
2y°=180°-60°
2y°=120°
y°=120°÷2
y°=60°
Hence, the value of y is 60°.
Thus, the correct option is D) 60°
f={(x,y)| y=x+3} find f(3.6) & f(-0.2)
Rectangle 800 feet long and 700 feet wide. if fencing costs $13 per yard, what will it cost to place fencing around the playground
Find the angular size of a circular object with a 3-inch diameter viewed from a distance of 4 yards.
Final answer:
To determine the angular size of an object with a diameter of 3 inches from 4 yards away, one can use the formula θ = d / D to calculate θ in radians, which is then converted to degrees, resulting in an angular size of approximately 1.19°.
Explanation:
To find the angular size of a circular object with a 3-inch diameter viewed from a distance of 4 yards, one can use the following formula for angular size θ (in radians): θ = d / D, where d is the diameter of the object and D is the distance to the object.
First, we convert the diameter and the distance to the same units. There are 36 inches in a yard, so 4 yards is 144 inches:
Diameter (d): 3 inchesDistance (D): 4 yards = 144 inchesThen, we calculate the angular size in radians:
θ = d / D = 3 inches / 144 inches = 0.0208333...
This can be converted to degrees by multiplying with 180/π:
θ(in degrees) = 0.0208333... * (180/π) = 1.19° (approximately).
Thus, the angular size of the object is approximately 1.19° when viewed from a distance of 4 yards.
Complete the multiplication: AB.
A.
B.
C.
D.
if a mixture of a 6% acid solution with an 11% acid solution is to be made, how much of each solution is needed to make 10 liters of an 8% acid solution
What is the value of theta , if tan theta = -1/squareroot3
The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. find the length of the median of the trapezoid.
Final answer:
The length of the median of the trapezoid with perpendicular diagonals of lengths 8 and 10 is 9 units. This is calculated using properties of the median and the Pythagorean theorem.
Explanation:
The question you've asked regarding the median of a trapezoid with perpendicular diagonals of lengths 8 and 10 can be resolved by recognizing a property of the median in a trapezoid. The median (also known as the mid-segment) of a trapezoid is parallel to the bases and its length is equal to the average of the lengths of the bases. Since the diagonals are perpendicular, they would form right triangles with the two bases and the median line, dividing the trapezoid into four right triangles.
Let's denote the lengths of the bases as a and b, and the median as m. We know the diagonals intersect at their midpoints, thus splitting each into segments of lengths 4 and 5. Now we can form two right triangles sharing the median as a common side. Applying the Pythagorean theorem, we get two equations: m² + 4² = a² for the first right triangle and m² + 5² = b² for the second right triangle.
Since the median is the average of the bases, we have m = (a + b) / 2. Using the equations above, after some algebraic manipulations, we find that a² - b² = 16. With more manipulation, eventually, we find that (a - b)(a + b) = 16, and, since m is the average, 2m = a + b. Hence, we derive that 2m - b = (a - b), which simplifies to give us m = 9. This gives us the length of the median of the trapezoid as 9 units.
mary and jolene are both growing tomatoes in thier backyards to sell.jolene had 3 times as many as mary.but birds had eaten 6 before she picked them all.for the season they have a total of 54 tomotoes.how many did mary grow?
Mary grew 15 tomatoes. A simple algebraic equation was formulated and solved to determine the quantity: m + 3m - 6 = 54, where m represents the number of tomatoes grown by Mary.
Given that Jolene had 3 times as many tomatoes as Mary and birds ate 6 of Jolene's tomatoes, and together they have 54 tomatoes, we need to set up an equation to solve for the number of tomatoes Mary grew.
Let's denote the number of tomatoes Mary grew as m. Therefore, Jolene grew 3m tomatoes. Since birds ate 6 of Jolene's tomatoes, we have 3m - 6 for the number of tomatoes left with Jolene. The equation representing the total number of tomatoes then becomes m (Mary's tomatoes) + (3m - 6) (Jolene's remaining tomatoes) = 54. Simplifying this equation:
m + 3m - 6 = 54
4m - 6 = 54
4m = 60
m = 15
Therefore, Mary grew 15 tomatoes.
The cost of a long-distance phone call is $0.46 for the first minute and $0.37 for each additional minute. if the total charge for a long-distance call is $6.75, how many minutes was the call?
1st minute = 0.46
additional minutes = 0.37
6.75 -0.46 = 6.29
6.29/0.37 =17
17 +1 = 18 total minutes
simplify (2 (radical 5) - 4) (3 (radical 5) +2)
Given the function f (x) = 10x + 4, find each of the following. f(8), f(-4), f (0)
f(8)=?
f(-4)=?
f(0)=?
If a line has a slope of 2 and contains the point (-2, 1), what is its equation in point-slope form?
At a grocery store, an uncooked beef roast is on sale for $4.99/lb. At the same store, prepared roast beef is at the deli for $2.99/100g. How many times more expensive is the deli roast compared to the uncooked roast?
Marijuana contains _____________ more carcinogens than cigarettes.
a. 20–30%
b. 40–60%
c. 50–70%
d. 60–80%
help plzzzzzzzz this is hard
Answer:
A is correct!
Step-by-step explanation:
Rachel has $100 in her savings account and deposits an additional $25 per week. joanna has $350 in her account and is saving $5 per week. after how many weeks will the two girls have the same amount of money?
The table below shows four systems of equations: System 1 System 2 System 3 System 4 4x − 5y = 2 3x − y = 8 4x − 5y = 2 10x − 7y = 18 4x − 5y = 2 3x − 8y = 4 4x − 5y = 2 10x + 3y = 15 Which pair of systems will have the same solution?
Answer:
System 1 and System 2 are equivalent.Step-by-step explanation:
The first and second system have the same solutions, the are equivalent systems of equations. Let's calculate solutions to demonstrate it:
System 1.[tex]\left \{ {{4x - 5y = 2} \atop { 3x - y = 8}} \right.[/tex]
If we multiply the second equations by -5, we can eliminate one variable and find the first solution:
[tex]\left \{ {{4x - 5y = 2} \atop { -15x +5y = -40}} \right.\\-11x=-38\\x=\frac{38}{11}[/tex]
Now, we use this value to find the other solution:
[tex]3x - y = 8\\3(\frac{38}{11})-y=8\\\frac{114}{11}-8=y\\ y=\frac{114-88}{11}=\frac{26}{11}[/tex]
The solution of the first system is [tex](\frac{38}{11} ;\frac{26}{11} )[/tex]
System 2.[tex]\left \{ {{4x - 5y = 2} \atop { 10x - 7y = 18}} \right.[/tex]
We do the same process than we did before, but this time we have to multiply by [tex]-\frac{5}{7}[/tex]:
[tex]\left \{ {{4x - 5y = 2} \atop { -\frac{50}{7}x + 5y = \frac{90}{7} }} \right.\\\frac{28x-50x}{7}=\frac{-90+14}{7}\\-22x=-76\\x=\frac{38}{11}[/tex]
Then,
[tex]4x - 5y = 2\\4(\frac{38}{11})-5y=2\\\frac{152}{11}-2=5y\\ 5y=\frac{152-22}{11}\\y=\frac{130}{5(11)}=\frac{26}{11}[/tex]
The solution of the second system is [tex](\frac{38}{11} ;\frac{26}{11} )[/tex]
Therefore, system 1 and system 2 are equivalent.
Find all solutions in the interval [0, 2π). 7 tan^3x - 21 tan x = 0
To solve this problem, the first thing we can do is to take out 7 tan (x) from the whole equation, this results in simplification:
7 * tan (x) * (tan (x)^2 - 3) = 0
Therefore the initial roots are taken from:
tan (x) = 0
Calculating for x:
x = 0, pi
The other roots can be taken from:
(tan (x)^2 - 3) = 0
Calculating for x:
tan (x)^2 = 3
tan (x) = ± sqrt (3)
x = pi/3 , 2pi/3 , 4pi/3 , 5pi/3
Therefore the solutions are:
0 , pi, pi/3 , 2pi/3 , 4pi/3 , 5pi/3