Answers
Answer:
Short leg: 9 in
Long leg: 40 in
Hypotenuse: 41 in
Step-by-step explanation:
This problem involves the Pythagorean Theorem, solving a system of equations, and solving a quadratic equation.
The Pythagorean Theorem applies because the question states this is a right triangle. That is one equation.
Two more equations come from descriptive relationships between various sides of the triangle.
Solving the system is done here with the Substitution Method, and solving the quadratic equation is done with factoring and the zero product property. Other methods could be used, but there is a character limit, so we can't dive into every way to solve this.
The Pythagorean TheoremPythagorean Theorem: [tex]a^2+b^2=c^2[/tex] where "c" is the hypotenuse of the right triangle.
"c" must be hypotenuse.
It doesn't matter which of the two legs' lengths is used for "a" & "b". However, because of other parts of the question, we'll need to know which one is which.
The question refers back multiple times to one specific leg (the short one). So, when we refer back to that leg in our equations, we need to make sure we're using the same leg each time.
Keeping track of which leg is which
Let's use "a" for the smaller leg, and "b" for the longer leg.
"c", as always, is the hypotenuse.
Setting up the relationships between the sidesNext, the question gives two statements relating the side lengths of the triangle to each other.
For the first statement, "the length of the longer leg of a right triangle" (that's "b") "...is 13 inches more than three times the length of the shorter leg ("a").
Translating the first sentence into math:
"the length of the longer leg of a right triangle" means "b""...is..." means "=""...13 inches more than..." means " +13 " (start with what's coming next, and then do the adding)"...three times..." means " 3* ""... the length of the shorter leg." means "a".[tex]b=3a+13[/tex]
Similarly from the second statement, we get: [tex]c=3a+14[/tex]
Combining this with the Pythagorean Theorem, we have a system of 3 equations, and 3 unknowns.
[tex]b=3a+13[/tex]
[tex]c=3a+14[/tex]
[tex]a^2+b^2=c^2[/tex]
Solving a system of equationsTo solve a system of equations, there are two main methods:
The substitution methodThe elimination methodEither method can be used, but the substitution method is the most intuitive.
Substitution method
To solve a system with the substitution method, isolate variable in an equation, written in terms of the other variables, and substitute them in to try to get a single equation with a single unknown.
Looking at our system, the first two equations already have "b" and "c" isolated in terms of "a". If we substitute both of these into the third equation, we'll have a single equation in terms of a, and can attempt to solve it.
[tex]b=3a+13[/tex]
[tex]c=3a+14[/tex]
[tex]a^2+b^2=c^2[/tex]
Substituting...
[tex]a^2+(3a+13)^2=(3a+14)^2[/tex]
Rewrite the square binomials as a product...
[tex]a^2+(3a+13)(3a+13)=(3a+14)(3a+14)[/tex]
Apply FOIL (a combination of the distributive property, the commutative properties of multiplication, and combining like terms)...
[tex]a^2+(9a^2+39a+39a+169)=(9a^2+42a+42a+196)\\a^2+(9a^2+78a+169)=(9a^2+84a+196)\\[/tex]
Observe that the equation is a degree-2 polynomial (polynomial with highest power of 2; aka quadratic). Move all terms to one side to get the equation equal to zero in preparation for solving:
[tex]a^2+(9a^2+78a+169)=(9a^2+84a+196)\\a^2+9a^2+78a+169-(9a^2+84a+196)=0\\a^2-6a-27=0\\[/tex]
There are a few ways to solve this, including the quadratic formula. However, factoring the polynomial and using the zero product property is quite efficient here:
Factoring to solve a Quadratic
Since the leading coefficient is 1, we can use the shortcut and find factors of -27 that add to make -6.
-27 has factor pairs of:
1 and -273 and -99 and -327 and -13 and -9 add to make -6, so factoring the polynomial is
[tex]a^2-6a-27=0\\(a+3)(a-9)=0[/tex]
Applying the zero product property...
[tex](a+3)=0[/tex] or [tex](a-9)=0[/tex]
So a=-3 or a=9. Given that this represents a length in a triangle, we discard the negative solution. So a=9.
Finding the other two unknowns
To finish up, substitute the "a" back into the two equations for "b" and "c".
[tex]b=3a+13\\b=3(9)+13\\b=27+13\\b=40[/tex]
[tex]c=3a+14\\c=3(9)+14\\c=27+14\\c=41[/tex]
The original problem set up our measurements in inches, so all three values are measured in inches.
Remember which length was which
"a": short leg
"b": longer leg
"c": hypotenuse
Short leg: 9 in
Long leg: 40 in
Hypotenuse: 41 in.
Related Questions
¿A qué porcentaje anual se han impuesto Q. 8000.00 que en 32 días han producido Q. 140.00? ¿ A qué porcentaje anual se han impuesto Q. 8000.00 que en 32 días han producido Q. 140.00 ?
Answers
Based on the amount being taxed, the period of taxation, and the tax, the annual percentage of tax is 19.96%.
What is the annual tax percentage?Assuming this rate was x, the following formula would apply:
Annual rate/ 365 days a year x number of days x amount to be taxed = Tax amount
Solving gives:
x/365 x 32 x 8,000 = 140
701.369863x = 140
x = 140 / 701.369863
= 19.96%
Find out more on tax rates at https://brainly.com/question/27865094?referrer=searchResults.
#SPJ1
Find the indicated side of the right triangle 60° 30° 3 x
Answers
Answer: [tex]3\sqrt{3}[/tex]
Step-by-step explanation:
[tex]\tan 60^{\circ}=\frac{x}{3}\\\\\sqrt{3}=\frac{x}{3}\\\\x=\boxed{3\sqrt{3}}[/tex]
What is the solution to the equation below? Round your answer to two decimal places. 3e2x + 12 = 42
Answers
Answer:
X= 5.52
Step-by-step explanation:
1. Solve the equation:
e×2x + 12= 42
x=15/e
2. Round x=15/e to the required place
x=5.52
Answer:
X=1.15
Step-by-step explanation:
Write the integral and solve for the area under the curve:
Answers
Answer:
Integral: [tex]\int\limits^\pi _0 {3sinx} \, dx[/tex]
Area: 6
Step-by-step explanation:
[tex]\int\limits^\pi _0 {3sinx} \, dx = -3[cosx]_{0} ^{\pi} = 6[/tex]
Mr. Harry had 4 daughters, each daughter had a brother. How many Children does Mr. Harry have?
Answers
Answer:
Mr. Harry has 5 children.... I know what you were going for...
Write a rule to describe each translation
Answers
Answer:
slide - translate
Step-by-step explanation:
you slided them both as neither of the figures changed
Find m angle angle KML
Answers
Answer:
52
Step-by-step explanation:
this is a triangle whose 2 sides are equalfirst take 180 which is the sum of interior angle minus 76 which is the angle givenyou will get 104 which is the other 2 anglesdivide by 2 since they are equal to get 52Kira plans to buy a used van that costs $14,000. The dealer requires a 20% down payment. The rest of the cost is
financed with a 4-year, fixed-rate amortized auto loan at 3.5% annual interest with monthly payments.
22
Complete the parts below. Do not round any intermediate computations. Round your final answers to the nearest cent
if necessary. If necessary, refer to the list of financial formulas.
(a) Find the required down payment.
S
▷
$ 0
(b) Find the amount of the auto loan.
Aa
(c) Find the monthly payment.
Answers
A: 2800
B: 11200
C: 250.39 per month
Which interval describes where the graph of the function is positive?
Answers
Answer:
D is the correct answer.
Step-by-step explanation:
Find the portion of the graph that is above the x-axis.
Answer:
D
Step-by-step explanation:
Coming in from - inf the graph is positive and remains so until yo get to x = -1 <==== then it is zero and continues to negative values
If you bought a stock last year for a price of $103, and it has risen 13.5% since then, how much is the stock worth now, to the nearest cent?
Answers
The stock worth now, to the nearest cent is $90.75.
How to find the current worth of stock?The current worth of stock = initial price of stock+increase in the price of stocks
If you bought stock last year for a price of $103, and it has risen 13.5% since then,
Let X be the worth of stock;
103 × 100 = (13.5% + 100%) × X
10300 = 113.5X
X = 10300/113.5
X = 90.74889868
X = $90.75
Hence, The stock worth now, to the nearest cent is $90.75.
Learn more about stock;
https://brainly.com/question/16553868
#SPJ1
Which of the following polynomials is in standard form?
A. F(x)=-3x+6x²-1+7x
B. F(x)=7x+6x² -1-3x4
C. F(x)=-3x4 -1+6x² +7x
D. F(x)=-3x +6x² +7x-1
Answers
Answer:
d
Step-by-step explanation:
Use the five numbers 12, 11, 19, 16, and 12 to complete parts a) through e) below.
a) Compute the mean and standard deviation of the given set of data.
The mean is x = and the standard deviation is s =
(Round to two decimal places as needed.)
Answers
Answer:
mean = 14 and standard deviation = 3.03
Step-by-step explanation:
mean = sum of all numbers ÷ by the numbers available
= (11 + 12 + 12 + 16 + 19) ÷ 5
= 70 ÷ 5
= 14
FOR STANDARD DEVIATION * USE A SCIENTIFIC CALCULATOR THEY WON'T PENALIZE YOU*
follow these steps:
1. press mode then STAT
2. then press 1-VAR
3. put all the numbers on the table in ascending order
4 after press AC
5. press shift then 1
6. you will see many things but we are only interested in standard deviation, press *Var*
7. then select the standard deviation sign and press the equal sign
To purchase 13700 worth of restaurant equipment for her business Maria made a down payment of 1500 and took out a business loan for the rest after 3 years of paying monthly payments of 371.16 she finally paid off the loan
What was the total amount Maria ended up paying for the equipment
How much internet did Maria pay on the loan
Answers
The total amount Maria ended up paying for the equipment will be $14,861.76. And The interest of Maria on the loan will be 8.48%.
What is Algebra?The analysis of mathematical representations is algebra, and the handling of those symbols is logic.
To purchase 13700 worth of restaurant equipment for her business.
Maria made a down payment of 1500 and took out a business loan for the rest, after 3 years of paying monthly payments of 371.16 she finally paid off the loan.
The total amount Maria ended up paying for the equipment will be
Total amount = 371.16 × 3 × 12 + 1500
Total amount = $14,861.76
The interest of Maria on the loan will be
Interset = [(14861.76 – 13700) / 13700] x 100
Interset = 8.48%
More about the Algebra link is given below.
https://brainly.com/question/953809
#SPJ1
5x -9=2y por suma y resta ecuacion
Answers
-45=2y
2y=-45
Y=-45/2
Ecuación:
5x-9=2y
-45=2y
2y=-45
Which expression is equivalent to (1−sinβ)(1+sinβ)/cos2β for all values of β for which (1−sinβ)(1+sinβ)/cos2β is defined?
Select the correct answer below:
tan2β
tanβ+secβ
tanβ
sec2β
1
Answers
Work Shown:
[tex]\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}\\\\\frac{1-\sin^2(\beta)}{\cos^2(\beta)}\\\\\frac{\cos^2(\beta)}{\cos^2(\beta)}\\\\1\\\\[/tex]
Therefore, [tex]\frac{(1-\sin(\beta))(1+\sin(\beta))}{\cos^2(\beta)}=1\\\\[/tex] is an identity for all values of β for which the original expression is defined. I used the difference of squares rule for the 2nd step. Then I applied the pythagorean trig identity for the 3rd step.
Please help !!! I will give 20 points to the correct answer!!
Answers
-2x =31 - 11
-2x = 20
X = 20/2
X = 10
Indicate the equation of the given line in standard form. Show all of your work for full credit.
The line containing the altitude to the hypotenuse of a right triangle whose vertices are P(-1, 1), Q(3, 5), and R(5, -5).
Answers
Answer: 24 sq units is the area of triangle.
Step-by-step explanation:
Given , (x1 , y1) = (-1, 1), (x2 , y2) = (3, 5) , (x3, y3) = (5, -5)
Using the Formula :
Area of triangle = [tex]\[\frac{1}{2}(x1(y2-y3) + x2(y3-y1) + x3(y1 -y2))\][/tex]
[tex]\[=\frac{1}{2}(-1(5-(-5)) + 3(-5-1) + 5(1-5)) \\=\frac{1}{2}(-10 -1 8 -20) \\=\frac{1}{2}(-48) = -24\][/tex]
Area cannot be negative. We only consider the numerical value of answer. Therefore, area of triangle = 24 sq units.
For more area of triangle related question:
https://brainly.com/question/14366937
#SPJ1
f of x is equal to 4x divided by the quantity of x minus 16 end quantity.
Answers
The zero of the equation is 0. Since degree of denominator is greater than degree of numerator, therefore, the horizontal asymptote is y=0
Asymptotes and zeros of a functionThe zero of the function is the point where the function is zero
Given the function below
f(x) = 4x/x-16
For the zeros
0 = 4x/x -16
4x = 0
x = 0
Hence the zero of the equation is 0
The vertical asymptote is the point where the denominator is zero
x - 16 = 0
x = 16
Since degree of denominator is greater than degree of numerator, therefore, the horizontal asymptote is y=0
Learn more on asymptotes and zeros here: https://brainly.com/question/16948935
#SPJ1
Lief starts saving for his future and deposits $425 each month at 5.7% annual interest, compounded monthly, for 25 years. At the end of 25 years, he has an account balance of $281,292.21. How much of this did Lief contribute and how much of this is interest?
Answers
Using proportions, it is found that the amounts are given as follows:
Amount deposited: $127,500.Amount of interest: $153,792.21.What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The amount deposited is of $425 each month for 25 years, that is, $425 in 25 x 12 times, that is:
D = 425 x 25 x 12 = $127,500.
The amount of interest is the total amount subtracted by the amount deposited, hence:
I = 281,292.21 - 127,500 = $153,792.21.
More can be learned about proportions at https://brainly.com/question/24372153
#SPJ1
Turkey sandwiches cost $2.50 and veggie wraps cost $3.50 at a snack stand. Ben has sold no more than $30 worth of turkey sandwiches and veggie wraps in the first hour of business. Let x represent the number of turkey sandwiches and y represent the number of veggie wraps. The inequality 2.50x+3.50y30 represents the food sales in the first hour.
If Ben has sold 4 veggie wraps, what is the maximum number of turkey sandwiches Ben could have sold?
Answers
The maximum number of turkey sandwiches Ben sold is 6.
What is the maximum number of turkey sandwiches Ben sold?Given this equation : 2.50x + 3.50y ≤ $30
If y is 4, take the following steps in order to determine the value of x:
Substitute for y in the above equation:
2.50x + (3.50 x 4) ≤ $30
2.50x + 14 ≤ $30
2.50x ≤ $30 - 14
2.50x ≤ $16
x ≤ 6.4
To learn more about inequality, please check: https://brainly.com/question/5031619
#SPJ1
Answer:
6
Step-by-step explanation:
Question 10: Each tablet in a bottle of 30 costs the pharmacy $4.60. The pharmacy has additional costs including supplies and shipping of $1.40 per pill. If a 12% profit is desired, how much would the pharmacy charge for entire bottle?
Answers
The pharmacy must charge for $201.6 for the entire bottle.
Profit is the financial gain to the seller after selling some commodity
Revenue is the total sale of the seller after selling some commodity
Expenses is the expenditure to the seller before selling the commodity
Profit = Revenue - Expenses
Cost of the tablet = $4.60
Total cost of the bottle=$4.60x30
=$138
Additional cost=$1.40 x 30
= $42
Total expenses=$180
Profit %=125
Revenue= $180 x (1+12/100)
=$180 x (112/100)
=$201.6
Therefore, The pharmacy must charge for $201.6 for the entire bottle for a profit of 12%
Learn more about profit and loss,
https://brainly.com/question/21637154
#SPJ10
Evaluate 3a-5b when a=-3 and b=-2
Answers
Answer: 1
Step-by-step explanation:
Substituting in the given values,
3a-5b=3(-3)-5(-2)=-9+10=
The length of a golden rectangle is 35 cm. Find the width to the nearest centimeter.
17cm
25cm
22cm
29cm
16cm
Answers
Answer:
22 cm
Step-by-step explanation:
First we need to know that The golden number
= (1+√(5))÷2
= 1.61803398875 (in decimal)
………………………………………………
We get the length of a golden rectangle
by multiplying its width by the golden number itself.
Then
The width = 35÷ 1.61803398875
=21.631189606245
(6.03)
What is the constant term in the expression 6x³y + 8x² + 5x + 7? (Input a numeric value only.)
Numerical Answers Expected!
Answer for Blank 1:
Answers
helppp im stuck :(
Hope you guys help me
Answers
Answer:
[tex]y=-\dfrac{1}{2}x+2[/tex]
Step-by-step explanation:
Define two points on the line:
[tex]\textsf{let}\:(x_1,y_1)=(0,2)[/tex]
[tex]\textsf{let}\:(x_2,y_2)=(4,0)[/tex]
Use the slope formula and the defined points to find the slope:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{0-2}{4-0}=-\dfrac{1}{2}[/tex]
Substitute the found slope and one of the points into the point-slope form of a linear equation:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-2=-\dfrac{1}{2}(x-0)[/tex]
Simplify:
[tex]\implies y-2=-\dfrac{1}{2}x[/tex]
[tex]\implies y=-\dfrac{1}{2}x+2[/tex]
What do large differences of the mean of each group indicate
Answers
The Actual Question:
What do large differences of the means of each group indicate? Select the correct answer.
A. The large differences do not indicate any Significant meaning in the given context.
B. The peppers with different weights aren't properly distributed between the two groups.
C. There is not enough information to analyze the differences.
D. The peppers with different weights are properly distributed between the two groups
Answer:
B. The peppers with different weights aren't properly distributed between the two groups.
What is a mean?
A quantity having a value intermediate between the values of other quantities; an average, especially the arithmetic mean.
The considerable discrepancies in the means of each group imply that: B. the peppers with varying weights are not adequately divided across the two groups.
So in the given situation above in the question, it can be concluded that large differences of the means from each group indicates that the peppers that are not properly distributed between the 2 different groups have different weights.
What is the volume of a regular pyramid having a base area of 24 inches and height of 6 inches
Answers
Answer:
V= 48 in^2
Step-by-step explanation:
Formula
Since the base has a known area, we do not need the full volume formula. This formula is
V = B * h / 3
B is the area of the base and h is the height measured from the top of the pyramid to the base meeting the base at right angles.
Givens
B = 24 in^2
h = 6 in
Solution
V = B * h / 3
V = 24 * 6/3
V= 48 in^2
evaluate 2/3 - 1/6 * 1/4
Answers
Answer: 0.625
Step-by-step explanation:
write the quadratic equation whose roots are -3 and 3, and whose leading coefficient is 5
Answers
Answer:
5x²- 1.8
Step-by-step explanation:
roots are -3 and 3, put them in brackets
(x - 3)(x + 3)
x² + 3x - 3x - 9
x² - 9
5x = -9
x = -1.8
5x² - 1.8
just to check:
5x² - 1.8
x² - 9
(x - 3)(x - 3)
x = -3 and 3.
1.9^4 × 10 × 2.9 x 10^1
Answers
3779.309 is the answer....