13.15.Verify that the six trigonometric function are well-defined. That is, show that it does not matter which right triangle with interior angle theta you choose- these six rayios will not change
Answers
We need to show that if we choose right triangle with the interior angle θ, the ratios of the sides will be the same as shown in below figure.
What is a trigonometry?The mathematical subject of trigonometry is the study of the connections between the angles and sides of triangles. It entails investigating trigonometric functions like sine, cosine, and tangent, which examine the relationship between a triangle's sides' lengths and its angles.
Let's consider a right triangle with acute angles α and β, as shown below:
To relate the sides of this triangle, we can use the Pythagorean theorem:
a² + b² = c²
Now, let's define the six trigonometric functions in terms of the sides of the triangle:
Sin(θ) = a/c
Cos(θ) = b/c
tan(θ) = a/b
csc(θ) = c/a
sec(θ) = c/b
cot(θ) = b/a
We want to show that these functions are well-defined, i.e., they do not depend on the particular triangle we choose. To do this, we need to show that if we choose another right triangle with the same interior angle θ, the ratios of the sides will be the same.
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Related Questions
Give a recursive definition for the following set of ordered pairs of positive integers ([tex]a|b[/tex] means that a is a factor of b): [tex]S=[/tex]{[tex](a,b)|a \in Z^+, b \in Z^+, a|b[/tex]}
Answers
A recursive definition for the set S of ordered pairs of positive integers (a,b) where a is a factor of b can be given as follows:
1. The ordered pair (1, b) is in S for all positive integers b.
2. If a is a factor of b, then (a, b) is in S if and only if (a, b/a) is in S.
The first rule establishes the base case for the recursion, stating that (1, b) is in S for all b. The second rule provides the recursive step, stating that (a, b) is in S if and only if (a, b/a) is in S, where b/a is an integer division of b by a, and a is a factor of b. This recursive definition ensures that all pairs (a, b) in S have a as a factor of b.
The monthly cost (in dollars) of a long-distance phone plan is a linear function of the total calling time (in minutes). The monthly cost for 39 minutes of calls is 18.45 and the monthly cost for 56 minutes is $20.66. What is the monthly cost for 50 minutes of calls?
Answers
Step-by-step explanation:
We can use the two data points given to find the equation of the line, which gives the monthly cost (y) in terms of the calling time in minutes (x).
First, we can find the slope of the line:
slope = (change in y) / (change in x)
slope = (20.66 - 18.45) / (56 - 39)
slope = 0.219
Next, we can use one of the data points and the slope to find the y-intercept (b). Let's use the data point (39, 18.45):
y - y1 = m(x - x1)
y - 18.45 = 0.219(x - 39)
y - 18.45 = 0.219x - 8.541
y = 0.219x + 9.909
So the equation for the monthly cost is y = 0.219x + 9.909.
To find the monthly cost for 50 minutes of calls, we plug in x = 50:
y = 0.219(50) + 9.909
y ≈ $21.44
Therefore, the monthly cost for 50 minutes of calls is approximately $21.44.
NEED HELP WITH THIS TRIG QUESTION
Answers
The distance from the ball to the center of the green (CD) is approximately 13.2 yards.
How to find distance using trignometric ratios?Determine the values given: Determine the known angles and distances that are involved in the issue.Choose the proper trigonometric function: To link the angle and the distances, select the proper trigonometric function (such as sine, cosine, or tangent) based on the given angle and the sides of the triangle created by the points.Use the trigonometric formula: Using the selected function and the unknown distance as the variable, write the trigonometric equation and replace the specified values .Calculate the distance: To determine the unknown distance on one side of the equation, use algebraic methods and according to the requirements of the problem, round the final response to the specified number of significant figures or decimal places.Given:
AB = 110 yards (distance from marker to center of the green)
BC = 45 yards (distance from marker to golfer's ball)
angle BAC = 115° (angle formed when golfer turns towards the ball)
To find CD (distance from ball to center of the green):
Apply the cosine function to angle BAC:
cos(115°) = AB/BC (∵cos θ = Adjacent side/ Hypotenuse)
Substitute the given values:
cos(115°) = 110/45
Calculate cos(115°):
cos(115°) ≈ -0.2924 (rounded to four decimal places)
Multiply both sides by BC to solve for AB:
AB = BC * cos(115°)
AB ≈ 45 * -0.2924
AB ≈ -13.16 yards (rounded to two decimal places)
Note: (The AB is negative, as the ball is moving in the opposite direction of the golfer's turn.)
CD = |AB| ≈ 13.16 yards (rounded to two decimal places)
CD ≈ 13.2 yards (rounded to one decimal point)
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Oliver is buying a new dishwasher. The dishwasher’s price is $285.10 after tax. He has a few payment options. He can put it on his debit card, which would take the money from his savings account. His savings account earns interest at a rate of 1.8% annually. He could also put it on one of two different credit cards. Card A has an annual interest rate of 9%, charged monthly, and charges a flat 1.2% fee on the initial value of the purchase (note: the fee accrues interest too). Card B has an annual interest rate of 12%, charged monthly, but no fee for purchases. Oliver thinks it will take five months for him to earn the additional money to offset the purchase. In all cases, the interest accrues according to this equation:
A = P(1 + r)n, where A is the final dollar amount, P is the principal (initial amount borrowed), r is the interest rate for each interest period, and n is the number of interest periods.
Answers
Oliver withdraws an amount of $285.10 from his saving account and the interest is $26.60
How to calculate the interest?Compound interest is a type of interest that is paid on the closing balance at the end of the previous year, which includes the interest paid in previous years.
His saving account earns 1.8% annually
The interest Oliver could have earned in five months:
Monthly interest = Annual interest ÷ 12
Monthly interest = 1.8% ÷ 12
Monthly interest = 0.018 ÷ 12 = 3/2000
A = P(1 + r)n
After five months = Principle × (1 + interest)ⁿ
After five months = 285.10 × (1 + 0.018)⁵
After five months = 311.70
Interest earned = 311.70 - 285.10 = $26.60
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Determine the amount thabang will receive if they decided to share their money according to hours
Answers
If Thabang and Ndivhu decided to share their money according to hours, then Thabang will receive R240.
Thabang worked for 3 hours and Ndivhu worked for 2 hours.
Together, they worked for a total of 5 hours. They were paid R400 for their work,
So, we calculate their hourly-rate by dividing the total amount by the total number of hours worked;
⇒ Hourly rate = (Total amount paid)/(Total number of hours worked),
⇒ Hourly rate = R400/5,
⇒ Hourly rate = R80 per hour,
Now, we find amount Thabang will receive if it is decided to share the money according to "hours-worked".
Thabang worked for 3 out of the total 5 hours, so he should receive 3/5 of the total amount paid,
⇒ Amount Thabang will receive = (Hours Thabang worked / Total hours worked) × Total amount paid,
⇒ Amount Thabang will receive = (3/5) × R400,
⇒ Amount Thabang will receive = R240,
Therefore, Thabang will receive R240 if they decided to share their money according to hours worked.
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The given question is incomplete, the complete question is
Thabang and his friend Ndivhu share a job of cleaning a neighbors yard and they were paid R400, Thabang worked 3 hours and Ndivhu worked for 2 hours. Determine the amount Thabang will receive if they decided to share their money according to hours.
VERY EASY 20 POINTS
The graph of y = f(x) is shown below, in red. Find the equation that corresponds to the blue graph.
Answers
Answer:
y = f(x) + 3
Hope this helps!
Step-by-step explanation:
Because the graph went up 3...
Answer:
y = f(x) + 3
Hope this helps!
Step-by-step explanation:
Because the graph went up 3...
It is given that M is the midpoint of and . Midpoints divide a segment into two congruent segments, so . Since and perpendicular lines intersect at right angles, and are right angles. Right angles are congruent, so . The triangles share , and the reflexive property justifies that . Therefore, by the SAS congruence theorem. Thus, because _____________. Finally, ΔPKB is isosceles because it has two congruent sides.
Answers
Complete paragraph proof would be detailed proof.
Given that M is the midpoint of PK and PK ⊥ MB, we need to prove that △PKB is isosceles.
Proof,
Since M is the midpoint of PK, PM ≅ KM.
Also, since PK ⊥ MB, we have ∠PMB and ∠KMB are right angles.
Since right angles are congruent, we have ∠PMB ≅ ∠KMB.
Now, by the SAS congruence theorem, we have △PMB ≅ △KMB because they share side MB, and PM ≅ KM and ∠PMB ≅ ∠KMB.
Thus, we have BP ≅ BK because corresponding parts of congruent triangles are congruent.
Therefore, △PKB has two congruent sides and isosceles by definition.
Hence, we have proven that △PKB is isosceles.
Correct Question :
Complete the paragraph proof. Given: M is the midpoint of PK PK ⊥ MB Prove: △PKB is isosceles It is given that M is the midpoint of PK and PK ⊥ MB. Midpoints divide a segment into two congruent segments, so PM ≅ KM. Since PK ⊥ MB and perpendicular lines intersect at right angles, ∠PMB and ∠KMB are right angles. Right angles are congruent, so ∠PMB ≅ ∠KMB. The triangles share MB, and the reflexive property justifies that MB ≅ MB. Therefore, △PMB ≅ △KMB by the SAS congruence theorem. Thus, BP ≅ BK because . Finally, △PKB is isosceles because it has two congruent sides.
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y = |x - 5| + |x + 5| if x >5
Answers
Answer:
y = 2x
Step-by-step explanation:
You want the simplified form of y = |x -5| +|x +5| if x > 5.
Turning pointsThe graph of the whole function will have turning points where the absolute value expressions are 0:
x -5 = 0 ⇒ x = 5
x +5 = 0 ⇒ x = -5
For values of x > 5, we are concerned with that portion of the graph that is to the right of both of these turning points. Hence, both absolute value expressions are positive and unchanged by the absolute value bars.
y = (x -5) +(x +5) . . . . . . if x > 5
y = 2x . . . . . . . . . . . . . . collect terms
The simplified function is y = 2x.
__
Additional comment
The attached graph shows y=2x and the given function for x > 5. They are identical. (The y=2x graph is shown dotted, so you can see the red graph of the given function.)
Write the polynomial in standard form.
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Answer:
[tex] - 9 {g}^{3} + {g}^{2} + 10g - 5[/tex]
CAN SOMEONE HELP WITH THIS QUESTION?
Answers
The area under the curve is 522 units².
What is the area under the curve?The area under the curve is calculated by integrating the function and finding the definite integra as shown below.
f(x) = 8x + 10
The integra of the function;
f(x)' = 4x² + 10x
Find the definite integra at the boundaries, x = 19 and x = 22
f(19)' = 4(19)² + 10(19)
F(19)' = 1,634
f(22)' = 4(22)² + 10(22)
F(22)' = 2,156
Area = 2,156 - 1,634 = 522 units²
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what is the average mass of the people in kg ?
Answers
Answer:
Step-by-step explanation:
To find the average mass of the six people, we need to divide the total mass by the number of people.
The total mass of the six people is 1/2 tonne, which is equivalent to 500 kg (since 1 tonne = 1000 kg).
So, the average mass of the six people is:
500 kg / 6 = 83.33 kg (rounded to two decimal places)
Therefore, the average mass of each person is approximately 83.33 kg.
Video Wa community into four adve graph shows the total number of each zone for a three-week period. 1,000 900 800 700 600 500 400 300 200 100 0 Number of Customers Zone 1 Video Warehouse Zone 2 Zone 3 A. 3:5 OB. 3:2 OC. 2:1 OD. 1:2 Zone 4 1. During the three weeks, how many customers came from Zones 3 and 4? A. between 900 and 1,000 B. between 1,000 and 1,100 C. between 1,100 and 1,200 D. between 1,200 and 1,300 2. Approximately what is the ratio of custo from Zone 1 to customers from Zone 3
Answers
The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.
1) During 3 weeks bound
Customers from zone 3 = 900
Customers from zone 4 = 320
So, total customers = (900 + 320)
= 1220
Hence, option D is correct i.e between 1200 and 1300.
2) Customer from zone 1 = 450
Customers from zone 3 = 900
So, Customer from zone 1 / Customer from zone 3 = 450/900
= 1/2
Hence, option 2 i.e., 1:2 is correct.
3) 3/5 pound of clay is used to make = 1 bowl
So, 1 bound of clay is used to make = 5/3 bowl
By unitary method,
10 pounds of clay is used to make = 10 *(5/3)
= 16.66
= 17 bowls (approx)
Hence, option d is correct answer.
Therefore, The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.
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NEED ASAP PLS
b. Dwayne makes 10 hours of long-distance calls in a month. How
much is his bill for that month?
Answers
if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.
How to determine how much is his bill for that monthThe cost of long-distance calls can vary depending on the service provider and the country being called. Without that information, it's difficult to give an accurate answer to this question.
Assuming a standard rate of $0.10 per minute for long-distance calls, we can calculate Dwayne's bill as follows:
10 hours = 600 minutes (since there are 60 minutes in an hour)
600 minutes x $0.10 per minute = $60
Therefore, if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.
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I have three math questions
Decide whether the given ordered pair is a solution to the system of linear inequalities
1 - y > x - 6
y < x -1
(5,2)
2 - y (< with a line under it i dont know how to type it) 2x
y (> with a line under it) x
(-3, -6)
3 - (1 over 2)x +3y < 8
y (> with a line under it) 1
(0, (2 over 3) )
Answers
- y > x - 6 ==> 2 > 5 - 6 ==> 2 > -1 (true)
- y < x - 1 ==> 2 < 5 - 1 ==> 2 < 4 (true)
Both inequalities are true when x=5 and y=2, so (5,2) is a solution to the system of linear inequalities.
2. To check if (-3,-6) is a solution, we need to substitute x=-3 and y=-6 into both inequalities and check if both inequalities are true:
- y ≤ 2x ==> -6 ≤ 2(-3) ==> -6 ≤ -6 (true)
- y ≥ x ==> -6 ≥ -3 (false)
The second inequality is not true when x=-3 and y=-6, so (-3,-6) is not a solution to the system of linear inequalities.
3. To check if (0, 2/3) is a solution, we need to substitute x=0 and y=2/3 into both inequalities and check if both inequalities are true:
- (1/2)x + 3y < 8 ==> (1/2)(0) + 3(2/3) < 8 ==> 2 < 8 (true)
- y > 1 ==> 2/3 > 1 (false)
The second inequality is not true when x=0 and y=2/3, so (0, 2/3) is not a solution to the system of linear inequalities.
Answer:
1. Not true, so (5,2) is not a solution
2. Not true, so (-3,-6) is not a solution.
3. Not true, so (0, [tex]\frac{2}{3}[/tex]) is not a solution.
Step-by-step explanation:
Substitute 5 for x and 2 for y
1 - y > x - 6
1 - 2 > 5 - 6
-1 > -1
This is not true. -1 is not greater than -1
y < x - 1
Substitute -3 for x and -6 for y
2 - y [tex]\leq[/tex] 2x
2 - (-6) [tex]\leq[/tex] 2(-3)
8 [tex]\leq[/tex] -6
This is not true. 8 is not less than -6.
y [tex]\geq[/tex] x
Substitute 0 for x and [tex]\frac{2}{3}[/tex]
3 - [tex]\frac{1}{2}[/tex] x + 3y < 8
3 - [tex]\frac{1}{2}[/tex] (0) + 3([tex]\frac{2}{3}[/tex]) < 8
3 - 0 + [tex]\frac{6}{3}[/tex] < 8
3 + 2 < 8
6 < 8
This is true. 6 is less than 8
y [tex]\geq[/tex] 1
[tex]\frac{2}{3}[/tex] [tex]\geq[/tex] 1
This is not true. [tex]\frac{2}{3}[/tex] is not greater than or equal to one.
The ordered pair has to be a solution for both equations for the ordered pair to be a solution for the system.
Helping in the name of Jesus.
Simplify.
8x^3/2 × -7x^2/3
And pls add an explanation. I know the answer but not how to get there.
Answers
The expression 8x^3/2 × -7x^2/3 can be simplified to give
-56 x^(13/6).How to simplify the expressionTo reduce the given expression, we can employ the principles of exponents and carry out the multiplication.
The given expression is:- 8x^(3/2) * (-7x^(2/3))
Utilizing this property, combining two terms with exponents entails adding the exponents when they feature the same bases. Thus, we establish:
8 * (-7) = -56 (coefficient multiplication)
x^(3/2 + 2/3) results in x^(13/6) (exponent addition)
Bringing it all together, we have:
8x^(3/2) * (-7x^(2/3)) = -56x^(13/6)
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URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Put the items in order from least steep to steepest rate of change
Answers
Answer:
1) Michele
2) f(t)
3) Elisa's puppy
4) g(x)
Step-by-step explanation:
This is because Michele's situation has a rate of change of -10, f(t) has -6.2, Elisa's puppy's situation has a rate of change of +5, and g(x) has 8.1
Hence,
-10 < -6.2 < 5 < 8.1
Hope this helps and be sure to mark this as brainliest! :)
How many liters each of a 50 % acid solution and a 65 % acid solution must be used to produce 60 liters of a 55 % acid solution? (Round to two decimal places if necessary.)
Answers
To produce 60 liters of a 55% acid solution, we would need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution.
Let number of liters of 50% acid solution needed be = "x", and
Let number of liters of 65% acid solution needed to produce 60 liters of a 55% acid solution. be = y,
To solve for x and y, we can use the following system of equations:
The "total-volume" of the solution is 60 liters,
⇒ x + y = 60, ...equation(1)
We know that, the total amount of acid in the solution is equal to the concentration of the final solution times its volume;
⇒ 0.50x + 0.65y = 0.55(60) ...equation(2),
On simplifying equation(2),
We get,
⇒ 0.50x + 0.65y = 33,
From equation(1), we have ⇒ x = 60 - y,
Now, we substitute "x" into the second equation and solve for y)
⇒ 0.50(60 - y) + 0.65y = 33,
⇒ 30 - 0.50y + 0.65y = 33,
⇒ 0.15y = 3,
⇒ y = 20
So, we need 20 liters of the 65% acid solution.
⇒ x + y = 60,
⇒ x + 20 = 60,
⇒ x = 40,
Therefore, we need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution to produce 60 liters of a 55% acid solution.
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The formula for the volume of a right circular cylinder is V=πr2h. If r=2b and h=5b+3, what is the volume of the cylinder in terms of b?
Answers
The volume of the cylinder in terms of b is expressed as: (20b³ + 12b²)π cubic units.
What is the Volume of a Cylinder?Where r represents the radius of a cylinder and h represents its height, the volume of the cylinder can be calculated using the formula:
V = πr²h.
Given the following:
Radius (r) = 2b
Height (h) = 5b + 3
Volume (V) = π * (2b)² * (5b + 3)
Volume (V) = π * 4b² * (5b + 3)
Volume (V) = π * 20b³ + 12b²
Volume (V) = (20b³ + 12b²)π cubic units
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Martina is learning different kinds of Latin dance she went to 6 salsa lessons and 12 rumba lessons. Each cost four dollars. let T equal the total number of lessons let c equal the total cost of classes .write an equation to find the total number of the lessons and ration to find the total cost of the lessons
Answers
The equation to find the total number of lessons is T = 6 + 12 = 18, and the ratio to find the total cost of the lessons is c:T = 72:18 or 4:1.
The equation to find the total number of lessons (T) can be written as:
T = number of salsa lessons + number of rumba lessons
Substituting the given values, we get:
T = 6 + 12 = 18
The ratio to find the total cost of the lessons (c) can be written as:
c = cost per lesson x total number of lessons
Substituting the given values, we get:
c = $4 x 18 = $72
Therefore, the equation to find the total number of lessons is T = 6 + 12 = 18, and the ratio to find the total cost of the lessons is c:T = 72:18 = 4:1.
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What is the range of the function in the graph?graph on the f-e axis, between the points (6, 40) and (12, 100)
A. 6≤e≤12
B. 40≤f≤100
C. 6≤f≤12
D. 40≤e≤100
Answers
For the function in the graph the range is option B: 40≤f≤100.
What is range of function?The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain. The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain.
The range of the function is the output values, or the y-coordinates of the function.
For the given graph, the output of the function is from 40 to 100 thus, the range is:
40≤f≤100
Hence, for the function in the graph the range is option B: 40≤f≤100.
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is x^3 an exponential function
Answers
The town's emergency response planning committee wants to place four emergency response centers at the four corners of town Each would serve the people who live within 3 mi of the response center Sketch the loci of points for the areas served What are the problems with this idea ? What is one potential solution ?
Answers
The town's emergency response planning committee has proposed to place four emergency response centers at the four corners of the town, with each center serving the people who live within 3 miles of the respective response center. The idea is to provide coverage to the entire town and ensure prompt emergency response for the residents. However, there are some problems with this idea.
Unequal coverage: Placing the emergency response centers at the four corners of the town may result in unequal coverage for the residents. Depending on the size, shape, and population distribution of the town, some areas may be farther away from the response centers, resulting in longer response times and reduced effectiveness in emergency situations.
Overlapping coverage: Placing four response centers in a small town may result in overlapping coverage areas, where the coverage areas of multiple response centers overlap with each other. This may lead to duplication of resources and inefficiencies in emergency response efforts.
Limited reach: Placing response centers only at the four corners of the town may result in limited reach for certain areas, especially those located in the middle or farther away from the corners. This may leave some residents outside the 3-mile coverage radius without access to timely emergency response services.
One potential solution to address these problems could be to use a more strategic approach to determine the locations of the emergency response centers. This could involve conducting a thorough analysis of the town's population density, geographical features, road network, and existing emergency resources. Based on this analysis, the response centers could be strategically placed at locations that provide the best coverage to the entire town, considering factors such as response time, resource allocation, and accessibility.
For example, instead of placing all the response centers at the corners of the town, they could be distributed more evenly across the town to ensure more equitable coverage. Additionally, the use of advanced GIS (Geographical Information System) technology and modeling techniques could help in identifying optimal locations for the response centers, taking into account various factors such as population density, road network, and travel time.
Furthermore, collaboration and coordination among the emergency response centers, along with proper communication and information sharing, can help in avoiding duplication of resources and improving the efficiency of emergency response efforts.
In conclusion, while the idea of placing four emergency response centers at the four corners of town may seem simple, there are potential problems such as unequal coverage, overlapping coverage, and limited reach. A more strategic and data-driven approach, considering factors such as population density, geographical features, and existing resources, can help in identifying optimal locations for the response centers and ensuring effective emergency response services for all residents of the town.
Find the measure of each indicated side. Round your answer to the nearest tenth.
Answers
Answer:
Step-by-step explanation:
By the law of cosines, [tex](59.7)^2=52^2+41^2-2\cdot 52 \cdot 41 \cos(\angle R).[/tex]
Thus, after solving for [tex]\cos(\angle R)[/tex], we get [tex]\cos(\angle R) = 0.1925211.[/tex] So taking the inverse cosine then yields [tex]\boxed{\angle R = 1.377 \text{ rad}}.[/tex]
Answer:
Step-by-step explanation:
By the law of cosines, [tex](59.7)^2=52^2+41^2-2\cdot 52 \cdot 41 \cos(\angle R).[/tex]
Thus, after solving for [tex]\cos(\angle R)[/tex], we get [tex]\cos(\angle R) = 0.1925211.[/tex] So taking the inverse cosine then yields [tex]\boxed{\angle R = 1.377 \text{ rad}}.[/tex]
Find the exact value of x.
Answers
Answer:
1) 13 **Explained below **
2) 4 **Explained below **
3) 6
4) 8
5) 15
6) 2.64575 **Explained below **
Step-by-step explanation:
All 6 problems can be solved using the Pythagorean theorem using the following two rules
If x is the hypotenuse and a, b are the shorter sides then
[tex]x = \sqrt{a^2 + b^2}[/tex]
If x is one of the shorter sides and a (or b) is the other shorter side and c is the hypotenuse then
[tex]x = \sqrt{c^2 - a^2\\[/tex] if a is the other shorter side
[tex]x = \sqrt{c^2 - b^2\\[/tex] if b is the other shorter side
I will just demonstrate for #1 , #2 and #6. You can figure out the reasoning behind the rest.
#1. x is the hypotenuse, 5 and 12 are legs(shorter sides)
[tex]x = \sqrt{5^2 + 12^2}\\\\x = \sqrt{25 + 144}\\\\x = \sqrt{169}\\\\x = 13\\\\[/tex]
# 2. x is one of the shorter sides (leg)
5 is the hypotenuse and 3 is the other shorter side
[tex]x = \sqrt{5^{2} - 3^{2}}\\\\x = \sqrt{5^{2} - 3^{2}}\\\\x = \sqrt{25 - 9}\\\\x = \sqrt{16}\\\\x = 4[/tex]
#6 hypotenuse is 4, short side = 3
[tex]x = \sqrt{4^{2} - 3^{2}}\\\\x = \sqrt{4^{2} - 3^{2}}\\\\x = \sqrt{16 - 9}\\\\x = \sqrt{7}\\\\x = 2.64575\\\\[/tex]
The price of a
home is $224,000. The bank requires a 20% down payment and three points at the time of closing. The cost of the home is financed with a 30-year fixed-rate mortgage at 7%. Complete parts (a) through (e) below.
Answers
(a) 20% of $224,000 is $44,800.
(b) 3 points on a $224,000 mortgage is $6,720.
(c) $44,800 + $6,720 is $51,520. So the total down payment and closing costs are $51,520.
(d) The mortgage amount is $224,000 - $51,520 = $172,480.
(e)
Monthly payment = $172,480 * (7%/12%)(30 years) = $1,234.15
Total interest paid over 30 years = $172,480 * 7% * 30 years = $243,720
Solve 9h2+9h=−2 by using the quadratic formula. Give an exact answer and simplify any fractions. If there are multiple answers, list them separated by a comma, e.g. 1,2. If there is no solution, enter ∅.
Answers
The value of h is ( -1/3, -2/3)
What is quadratic equation?Quadratic equation is a type of equation in which the highest power of variable is 2. A quadratic equation is represented as ax²+bx + c = 0
Solving, 9h²+9h = -2
9h²+9h+2 = 0
9h² +6h +3h +2 = 0
(9h²+6h) ( 3h +2) = 0
3h( 3h + 2)+1 ( 3h+2) = 0
(3h+2)(3h+1) = 0
Therefore ;
3h+2 = 0
3h = -2
h = -2/3 or
3h+1 = 0
3h = -1
h = -1/3
therefore the value of h is ( -1/3, -2/3)
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Joel has $100 more than Mike. After Joel gave half of his money to Mike, Mike had $500 more than Joel. How much did they have altogether?
Answers
ANSWER:
They have $1,100 altogether.
EXPLANATION:
- Originally, Mike had $500, while Joel had $600. That adds up to $1,100, and it makes Joel have $100 more than Mike.
- Half of Joel’s $600 is $300, which he gave to Mike. That makes Joel now have $300 himself.
- Adding $300 to Mike’s $500 is $800, which means Mike now has $800.
- $800 (Mike’s new amount) minus $300 (Joel’s new amount) is $500, which works because Mike now has $500 more than Joel.
-$300 + $800 is, of course, still $1,100.
A small college has 10 professors in the Mathematics Department. The department teaches pure math, applied math, and
statistics. Professors are apportioned using Hamilton's apportionment method, according to the number of majors in each
field. There are 4 pure math majors, 12 applied math majors, and 12 statistics majors.
If the department receives a grant to hire 1 more professor, will the Alabama paradox occur? Why or why not?
Answers
Answer:
Yes, because while the total number of professors increases, the number of pure math professors decreases.
Step-by-step explanation:
Using Hamilton's method:
pure math 4/28 = 0.142857
applied math 12/28 = 0.42857
statistics = 12/28 = 0.42857
10 × 0.142857 = 1.42857
10 × 0.42857 = 4.2857
10 × 0.42857 = 4.2857
numbers of professors:
pure math 1
applied math 4
statistics 4
total 9
Add 1 to pure math
Final original numbers of professors using Hamilton's method
pure math 2
applied math 4
statistics 4
total 10
Add 1 professor to department.
11 × 0.142857 = 1.5714
11 × 0.42857 = 4.71428
11 × 0.42857 = 4.71428
numbers of professors:
pure math 1
applied math 4
statistics 4
total 9
Add 1 to applied math and 1 to statistics
Final new numbers of professors using Hamilton's method
pure math 1
applied math 5
statistics 5
total 11
Despite the addition of 1 professor to the department, the field of pure mathematics went from 2 professors to 1 professor. This is an example of the Alabama paradox.
Answer: Yes, because while the total number of professors increases, the number of pure math professors decreases.
Add. (2x^4+c^3-4^2+3)+(4x^4-2x^3-x^2+10x+2)
Answers
The value of (2x^4+x^3-x^2+3)+(4x^4-2x^3-x^2+10x+2) is 6x⁴-x³-2x²+10x+5
What is addition of expression?Expression is a combination of numbers, variables, functions (such as addition, subtraction, multiplication or division etc.)
Adding two expressions together requires to collect like terms. like terms are terms with similar variable. For example, in 2x²+6x+x² , x² and 2x² are like terms.
Therefore, 2x⁴+x³-x²+3 + 4x⁴-2x³-x²+10x+2 can be simplified as;
2x⁴+4x⁴+x³-2x³-x²-x²+10x+2+3
= 6x⁴-x³-2x²+10x+2
therefore, the value of 2x⁴+4x⁴+x³-2x³-x²-x²+10x+2 is 6x⁴-x³-2x²+10x+5.
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Which of the following expressions is equal to 2?
4 x (one-half x 6) ÷ 3
6 ÷ (one-fourth x 3 x one and one-fourth)
5 x (one-third x 6) ÷ 5
10 − (one-fifth x 10) + 1
Answers
The expression is equal to 2 is 5 x (one-third x 6) ÷ 5. The correct option is C.
What is a mathematical expression?A mathematical expression is the collection of mathematical symbols that results from the proper combination of numbers and variables using operations like addition, subtraction, multiplication, division, exponentiation, and other as-yet-unlearned operations and functions.
Can be simplified each of the expressions see if them equals 2:
4 x (one-half x 6) ÷ 3 = 4 x 3 ÷ 3 = 4
6 ÷ (one-fourth x 3 x one and one-fourth) = 6 ÷ (3/4 x 5/4) = 6 ÷ (15/16) = 96/15 ≠ 2
5 x (one-third x 6) ÷ 5 = 5 x 2 ÷ 5 = 2
10 − (one-fifth x 10) + 1 = 10 - 2 + 1 = 9 ≠ 2
Therefore, the correct option is c. 5 x (one-third x 6)
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