Question 10The table below shows the amount of water that is in a bathtub that is being filled.Minutes1234Gallons ofWater12172227Which of the following functions could model the amount of gallons of water, f(x), that the bathtub will contain after x minutes of being filled?Аf(x) = 5x + 7Bf(x) = 5x + 12сf(x) = 7x + 5Df%) = 12x + 5No
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if the sum of the first three terms of a GP is half it's sum to infinity. find it's positive common ratio.
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Answer:
See below
Step-by-step explanation:
Sum of n = inf = a / ( 1-r) one half of this = 1/2 ( a/(1-r) )
Sum of first 3 terms = a ( 1- r^n) / ( 1-r) = a (1-r^3) / (1-r)
The underlined value are equal ( given in the Q)
1/2 a / (1-r) = a ( 1-r^3) / (1-r) Multiply both side by ( (1-r) )
1/2 a = a ( 1-r^3) divide both sides by a
1/2 = 1- r^3 subtract 1 from both sides
-1/2 = - r^3
1/2 = r ^3
r = cuberoot ( 1/2) = 1/2 ^(1/3) = .793700526
The equation of a curve is xy squared minus 2x square y squared equal 0. Find the gradient of the tangent to the curve at (1,2)?
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Solution
[tex]xy^2-2x^2y^2=0[/tex]we need to find the derivative with respect to x. dy/dx
[tex]\begin{gathered} \frac{d}{dx}(xy^2-2x^2y^2=0)=\frac{d}{dx}(xy^2)-2\frac{d}{dx}(x^2y^2)=0 \\ \\ \text{ using product rule} \\ \\ \Rightarrow x\frac{d}{dx}(y^2)+y^2\frac{d}{dx}(x)-2x^2\frac{d}{dx}(y^2)+2y^2\frac{d}{dx}(x^2)=0 \\ \end{gathered}[/tex]Applying chain rule
[tex]\begin{gathered} x\cdot\frac{d}{dy}(y^2)\cdot\frac{dy}{dx}+y^2-2x^2\cdot\frac{d}{dy}(y^2)\cdot\frac{dy}{dx}+2y^2(2x)=0 \\ \\ \Rightarrow x(2y)\frac{dy}{dx}+y^2-2x^2(2y)\frac{dy}{dx}+4xy^2=0 \\ \\ \Rightarrow2xy\frac{dy}{dx}+y^2-4x^2y\frac{dy}{dx}+4xy^2=0 \\ \\ \Rightarrow(2xy-4x^2y)\frac{dy}{dx}=-y^2-4xy^2 \\ \\ \Rightarrow\frac{dy}{dx}=\frac{-(y^2+4xy^2)}{2xy-4x^2y}=\frac{y^2+4xy^2}{4x^2y-2xy} \\ \\ \Rightarrow\frac{dy}{dx}=\frac{y^{2}+4xy^{2}}{4x^{2}y-2xy} \end{gathered}[/tex]At point (1,2)
[tex]\frac{dy}{dx}=\frac{(2)^2+4(1)(2)^2}{4(1)^2(2)-2(1)(2)}=5[/tex]Using slope intercept equation
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \\ y-2=5(x-1) \\ \\ y-2=5x-5 \\ \\ \Rightarrow y=5x-5+2 \\ \\ \Rightarrow y=5x-3 \end{gathered}[/tex]Assuming the pattern continues, what is S12 for the series −5 − 14 − 23 − 32 − …?A)-104B)-624C)-654D)-663
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The sum of the first 12 terms is s(12) = -624 if the sequence is series −5 − 14 − 23 − 32 option (B) is correct.
What is a sequence?
It is defined as the systematic way of representing the data that follows a certain rule of arithmetic.
It is given that:
The sequence is:
−5 − 14 − 23 − 32 −
s(12) means the sum of the 12 terms
s(12) = (12/2)[-5 + (12-1)(-9)]
s(12) = 6[-5 -99]
s(12) = 6[-104]
s(12) = -624
Thus, the sum of the first 12 terms is s(12) = -624 if the sequence is series −5 − 14 − 23 − 32 option (B) is correct.
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A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted. After 3 years, the population had grown to 270 wolves. If the population grows exponentially, then at what annual rate is the population growing? Round the answer to the nearest tenth of a percent.
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The population is growing at an annual rate of 16.2%
Here, we want to calculate the exponential growth rate
Mathematically, we can write the exponential equation of growth as follows;
[tex]\begin{gathered} P=I(1+r)^t \\ \\ \end{gathered}[/tex]Where P is the population after a certain number of years ( 270 after 3 years
I is the initial popultaion which is 200
r is the percentage we want to calculate
t is the number of yeats to reach P which is 3 in this case
[tex]\begin{gathered} 270=200(1+r)^3 \\ (1+r)^{3\text{ }}\text{ = }\frac{270}{200} \\ \\ (1+r)^3\text{ = 1.35} \\ \\ 1\text{ + r = }\sqrt[3]{1.35} \\ \\ 1\text{ + r = 1.162} \\ \\ r\text{ = 1.162 - 1} \\ \\ r\text{ = 0.162} \end{gathered}[/tex]To the nearest tenth of a percentage, this is 16.2%
What is 4 square root of 4?
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Answer:
8
Step-by-step explanation:
4[tex]\sqrt{4}[/tex]
The [tex]\sqrt{4}[/tex] is 2
4x2 = 8
Answer:The square root of 4 is simply just 2 or in other words, 4 = 2.
Step-by-step explanation:
The value of root 4 is equal to exactly 2. But the roots could be positive or negative or we can say there are always two roots for any given number.
how do I find the given z score
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The formula for calculating a z-score is
[tex]z=\frac{x-\mu}{\sigma}[/tex]Where:
*x is the raw score
*μ is the population mean
*σ is the population standard deviation
Given the following information, which is the best description of the data?Range—5, from 12 to 17Mode—14Median—14.5Mean—15The data is around 13.The data is around 12.The data is around 15.The data is around 17.
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we have that
the best description of the data is
The data is around 15A bag of cookies weighs 19.1 ounces. The bag contains 50 cookies. How much does each cookie weigh?
O2.6 ounces
O 0.3 ounces
O 0.382 ounces
O 3.82 ounces
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Answer: 0.382 ounces
Step-by-step explanation:
To find the unit rate of the weight of a single cookie, we will use division.
19.1 ounces per bag / 50 cookies in the bag = 0.382 ounces per cookie
To check our work, we will multiply 0.382 ounces per cookie by 50. Since the bag, with 50 cookies, weighs 19.1 ounces, 0.382 times 50 should equal about 19.1 ounces.
0.382 ounces * 50 cookies = 19.1 ounces ✓
17) A brand of coconut oil is updating its packaging. The old rectangular container was 1 foottall, 4 inches long and 2.5 inches wide, their new cylindrical container has a 4.5 centimeterradius and is 30 centimeters tall.a) Determine the volume of the rectangular container, 3 points(if needed round the final answer to the nearest hundredth.)b) Determine the volume of the cylindrical container. 3 points(if needed round the final answer to the nearest hundredth.)c) The company charges the same amount for each container. Is one of the containers abetter buy? Explain and show work that justifies your answer. 3 points
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a) Volume of rectangular container:
1 foot = 12 inches
[tex]\begin{gathered} V=\text{tall}\times long\times wide \\ V=12\times4\times2.5 \\ V=120inches^3 \end{gathered}[/tex]b) Volume of the cylindrical container:
[tex]\begin{gathered} V=\pi\times r^2\times h \\ V=\pi\times(4.5)^2\times30 \\ V=\pi\times20.25\times30 \\ V=607.5\pi \\ V=607.5(3.14) \\ V=1907.55\operatorname{cm}^3 \end{gathered}[/tex]c) First, we convert the volumes of the containers into the same units.
For volume of the rectangular container in centimeters:
1 inch3 = 16.3871 cm3
[tex]V=120\times16.3871=1966.45\operatorname{cm}^3[/tex]With this, we can say that the rectangular container has a larger volume than the cylindrical container. Therefore, the cylindrical container is a better buy.
Answer:
a) V = 120 in^3
b) V = 1907.55 cm^3
c) Cylindrical container is a better buy
Z divided 12 equal 9
Answers
Answer:
108
Step-by-step explanation:
12×9=108
just for the extra characters
Answer:
108
Step-by-step explanation:
z/12 = 9
z = 9 × 12
z = 108
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Select the vector along which a translation of the plane would map point A to its image T(A).
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The vector along which a translation of the plane would map point A to its image T(A) is: C. vector MN.
The types of transformation.In Geometry, there are different types of transformation and these include the following:
ReflectionDilationRotationTranslationWhat is a translation?A translation can be defined as a type of transformation which moves every point of the object in the same direction, as well as for the same distance.
In this context, we can reasonably infer and logically deduce that vector MN is a translation of the given plane which would map point A to its image point T(A) because they both move in the same direction.
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1. Fiber-Brite could clean 48 rugs in 12 hours. At that rate, how many rugs could Fiber-Brite
clean in 20 hours?
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Determine the odds against rolling a 4 on one roll
of a die.
Solution
P(4)=1
6
5
P (fails to roll a 4)=-
6
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Answer:
1/6
Step-by-step explanation:
SIX possible rolls 1,2,3,4,5,6
4 is 1 out of 6 or 1/6
Jacob is planting shrubs for a landscaping client. He planted more shrubs in the additional region so that the density of shrubs in the new region is equal to the density of shrubs in the rectangular region as shown.
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ANSWER:
52 shrubs
STEP-BY-STEP EXPLANATION:
We know the number of trees planted in a given area, so to calculate the number of total trees, we must calculate the area of the entire figure and thus calculate by means of a proportion
We calculate the total area if we divide the figure in two, as follows:
The area of zone A, which is a rectangle, is calculated as follows:
[tex]\begin{gathered} A_A=b\cdot h \\ A_A=50\cdot20 \\ A_A=1000 \end{gathered}[/tex]The area of zone B, which is a triangle, is calculated as follows:
[tex]A_B=\frac{b\cdot h}{2}[/tex]To calculate the value of the height of the triangle, we apply the Pythagorean theorem on the right triangle that is formed:
Where the hypotenuse is 25 meters and the other leg is 15 meters, just like this:
[tex]\begin{gathered} h^2=a^2+b^2 \\ 25^2=15^2+b^2 \\ b=\sqrt[]{25^2-15^2} \\ b=\sqrt[]{400} \\ b=20 \\ \text{therefore the height is 20 meters, replacing:} \\ A_B=\frac{30\cdot20}{2} \\ A_B=300 \end{gathered}[/tex]Now, the total area would be the sum of both areas:
[tex]\begin{gathered} A_T=1000+300 \\ A_T=1300 \end{gathered}[/tex]Which means that if in an area of 400 square meters (20 * 20) they can plant 16 shrubs, in an area of 1300 square meters they would be:
[tex]\begin{gathered} \frac{16}{400}=\frac{x}{1300} \\ \text{ solving for x} \\ x=\frac{16\cdot1300}{400} \\ x=52 \end{gathered}[/tex]In other words, a total of 52 shrubs can be raised throughout the region.
There are about 3.79 liters in 1 gallon. Kendra’s fish tank contains 100 gallons of salt water. About how many liters does the tank hold? Round your answer to the nearest whole number. Answer to get 60 points plus brainliest!
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Answer:
379 liters
Step-by-step explanation:
3.79 L / gal * 100 gal = 379 Liters
Kendra's fish tank can hold 379 liters of water under the given conditions.
Given that,
There are about 3.79 liters in 1 gallon. Kendra’s fish tank contains 100 gallons of salt water.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
3.79 liters = 1 gallon,
Now,
in the fish tank
100 gallon of water = 3.79× 100 liters
= 379 liters
Thus, Kendra's fish tank can hold 379 liters of water under the given conditions.
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Sammy is a baker who wants to make profit on his baked goods if it fits him 3.20 or o make a dozen cupcakes how much should he sell them if he wants to make 35% profit
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For there to be a profit,
[tex]\begin{gathered} \\ \text{profit}=\text{ selling price - cost price} \end{gathered}[/tex]From the question,
[tex]\begin{gathered} \text{Cost price = \$3.20} \\ \text{Percentage profit = 35\%} \end{gathered}[/tex]Concept: The formula for the percent profit is given below as
[tex]\text{percentage profit = }\frac{\text{profit}}{\cos t\text{ price}}\times100\text{ \%}[/tex]Emery borrowed money from her brother to buy a new phone and is paying off a fixed amount each week. After 2 weeks, she will owe $456, and after 5 weeks, she will owe $228.
a. What was the original amount Emery borrowed?
b. How much does she pay each week?
c. How useful are equations in point-slope and slope-intercept forms for answering each question?
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The $456 and $228 Emery owed from the amount she borrowed to buy a new phone after 2 weeks and 5 weeks gives;
a. The original amount owed is $608
b. Emery pays $76 each week
c. The point and slope and slope and intercept both gives the amount Emery pays each week as the slope, while the slope and intercept form further gives the original amount borrowed as the y–intercept, c
What is a straight line equation?A straight line equation is one which expresses a linear relationship between variables
Let y represent the original amount of money Emery borrowed from her brother, and let m represent the amount she pays each week to settle the loan, we have;
The given parameters are;
The amount Emery owes after 2 weeks = $456
The amount Emery will owe after 5 weeks = $228
a. The above information can be expressed by writing the following equations;
456 = y - 2•m...(1)
228 = y - 5•m...(2)
Subtracting equation (2) from equation (1) gives;
y - 2•m - (y - 5•m) = 456 - 228 = 228
3•m = 228
Therefore;
m = 228 ÷ 3 = 76
m = 76
The original amount Emery borrowed, y, from equation (1) is therefore;
y = 456 + 2•m
y = 456 + 2 × 76 = 608
The original amount Emery borrowed, y, is $608
b. The amount Emery pays each week, m = $76
c. The straight line equation in point and slope form is presented as follows;
[tex] y_2 - y_1 = m•(x_2 - x_1) [/tex]
From the above equation, the two points (456, 2) and (228, 5) given in the question can be used to find the slope, m, which is the amount paid each month
The slope and intercept form of a straight line equation is y = m•x + c
From the slope and intercept form, we have;
The slope, m = The amount Emery pays per month
The intercept, c = The original amount Emery borrowed
Therefore, both forms of the straight line equation gives the amount Emery pays each week, while the slope and intercept form also gives the original amount Emery borrowed
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Select the conic section that represents the equation.3x² + 3y²-2x + 4 = 0
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Given:
[tex]3x^2+3y^2-2x+4=0[/tex]First, we arrange the equation:
[tex]\begin{gathered} 3x^{2}+3y^{2}-2x+4=0 \\ 3x^2-2x+3y^2+4=0 \\ (3x^2-2x)+3y^2=-4 \end{gathered}[/tex]Further simplification of the given equation will give us a result with an imaginary number, making the equation a nonreal circle, or not a conic section.
help me pleaseeeeeeeeeeeeeeeeeeeeeeeee !!!
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Answer:
[tex]f(x)=\frac{5}{6}x+21[/tex]
Step-by-step explanation:
Slope = 5/6
Point = (-18,6)
Using Point-Slope form
y-6=5/6(x+18)
y-6=5/6x+15
y=5/6x+21
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How do I solve this problem? To find the distance AB across a river, a distance BC=250 is laid off on one side of the river. It is found that B =119 degrees and C=27 degrees. Find AB
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question shown in picture below!!(i only need the 2nd one)
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Given the following sequence:
[tex]\mleft\lbrace17,14,11,8,\ldots\mright\rbrace[/tex]notice that the common difference between each term is -3, therefore, we can use the explicit formula for a sequence:
[tex]\begin{gathered} a_n=a_1+(n-1)\cdot d \\ d=-3 \\ a_1=17 \\ \Rightarrow a_n=17+(n-1)(-3) \\ \Rightarrow a_n=17-3n+3=20-3n \\ a_n=20-3n \end{gathered}[/tex]we have that the explicit formula is a_n=20-3n, while the recursive formula is:
[tex]\begin{gathered} a_n=a_{n-1}+d \\ d=-3 \\ \Rightarrow a_n=a_{n-1}-3 \end{gathered}[/tex]You go on a hayride to take photographs of landmarks. The map shows your path and two landmarks. Each unit in the coordinate plane corresponds to 10 yards. Approximate your minimum distance from the giant pumpkin. If necessary, round your answer to the nearest tenth.
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Using the distance between a point and a line and the conversion of the units to yards, it is found that the minimum distance from the giant pumpkin is of 8.9 yards.
What is the distance between a points and a line?Suppose that we have a linear function defined according to the following rule, in standard notation:
Ax + By + C = 0.
And a point with coordinates given by:
P(x*,y*)
The shortest distance between the line and the point is given by:
[tex]d = \frac{|Ax^\ast + By^\ast + C|}{\sqrt{A^2 + B^2}}[/tex]
The line of the path in this problem has:
Intercept of 0, as when x = 0, y = 0.Slope of 0.5, as when x increases by 4, y increases by 2.Hence:
y = 0.5x.
-0.5x + y = 0
-x + 2y = 0.
The coefficients are:
A = -1, B = 2.
The giant pumpkin has coordinates given by:
(x*, y*) = (-4, -3).
Hence the distance in units is given by:
d = |(-1)(-4) + 2(-3)|/sqrt(5)
d = 2/sqrt(5)
d = 0.89 units.
Each unit is equivalent to 10 yards, hence the shortest distance in yards is given by:
0.89 x 10 = 8.9 yards.
What is the missing information?The problem is given by the image at the end of the answer.
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Factor completely x^2+10x+21
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Answer:
(x+3)(x+7)
Step-by-step explanation:
x2+10x+21
The middle number is 10 and the last number is 21.
Factoring means we want something like
(x+_)(x+_)
We need two numbers that
Add together to get 10
Multiply together to get 21
3 and 7:
3+7 = 10
3*7 = 21
Fill in the blanks in
(x+_)(x+_)
with 3 and 7 to get...
(x+3)(x+7)
Answer: (x+7)(x+3)
Step-by-step explanation:
We need to values that we will add and get 10 and two values we will multiply and get 21 which are 7 and 3
Using P.E.M.D.A.S, how do I solve this?
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Answer: 7
Step-by-step explanation:
First, do parenthesis- 2x4 is 8 and 8 times - is -8. Now it is 30/6+10-8. 30 divided by 6 is 5. 10-8 is 2. 5+2 is 7.
Write TWO facts about Pi
Answers
Solution:
Two facts about pi π
1) It is an irrational number.
2) It is ancient.
The ratio of two cars, the Thunderbolt and the Flash, in terms of their miles per gallon was 6:5. The new version of the Flash added 14 miles per gallon and now the ratio is 24:27 .
What is the miles per gallon of the Thunderbolt?
The Thunderbolt gets
miles per gallon.
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If the ratio the Thunderbolt and the Flash, in terms of their miles per gallon was 6:5 and the new version of the Flash added 14 miles per gallon and now the ratio is 24:27, then the miles per gallon of the Thunderbolt is 15.27 miles per gallon
The ratio the Thunderbolt and the Flash, in terms of their miles per gallon = 6:5
Then the miles per gallon of the Thunderbolt and the Flash is 6x and 5x respectively
The new version of the Flash added 14 miles per gallon
New ratio of the cars = 24:27
Therefore the equation is
5x+14 = 27x
27x-5x = 14
22x = 14
x = 7/11
The miles per gallon of the Thunderbolt = 24x
= 24×7/11
= 15.27 miles per gallon
Hence, If the ratio the Thunderbolt and the Flash, in terms of their miles per gallon was 6:5 and the new version of the Flash added 14 miles per gallon and now the ratio is 24:27, then the miles per gallon of the Thunderbolt is 15.27 miles per gallon
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Solve g(7) given g(X)= -3x+1
Answers
Step-by-step explanation:
that means we need to calculate the function result with x having a specific value.
for this we replace any appearance of x in the function expression by that specific value, and then we calculate.
g(x) = -3x + 1
g(7) means x = 7, and that gives us
-3×7 + 1 = -21 + 1 = -20
so, g(7) = -20
Answer:
-20
Step-by-step explanation:
Substitute 7 into the function g(x)
Since negatives "rule" positives,
7 * -3 = -21,
-21 + 1 = -20
Mrs. Ojo is 5 times older than her son. 3 years ago the product of their ages was 185. How old are they now?
need it now please.
Answers
(A – 3)(B – 3)= 185
AB – 3A – 3B + 9= 185
AB – 3(A + B)= 185 – 9= 176
AB – 3(A + B)= 176 •••> (2)
Put (1) in (2),
5B(B) – 3(5B + B)= 176
5B² – 3(6B)= 176
5B² – 18B – 176= 0
(B – 8)(B + 22/5)= 0
B= 8 and B= –22/5
When B= 8, from (1),
A= 5(8)= 40.
Thus, Mrs Ojo is 40 while her son is 8.
I have to send you the question
Answers
The given expression fro the sum of n terms:
[tex]\begin{gathered} S_n=\frac{n}{2}(a_1+a_n) \\ \text{ Where: }a_1\text{ is the firm term \& }a_n\text{ is the last term} \end{gathered}[/tex]From the given question we have:
[tex]a_1=8,a_n=79,\text{ n =6}[/tex]Substitute these value in the expression of sum of n terms
[tex]\begin{gathered} S_n=\frac{n}{2}(a_1+a_n) \\ S_6=\frac{6}{2}(8+79) \\ S_6=3(87) \\ S_6=3\times87 \\ S_6=261 \end{gathered}[/tex]So, sum of 6 terms is 261
in a 30 16 90° triangle given the short leg equals five find the long leg of the triangle
Answers
In a 30-60-90 triangle:
[tex]\begin{gathered} Hypotenuse_{\text{ }}=_{\text{ }}2\cdot short_{\text{ }}length \\ Long_{\text{ }}length=\sqrt[]{3}\cdot short_{\text{ }}length \\ \end{gathered}[/tex]so:
[tex]\begin{gathered} Long_{\text{ }}length=\sqrt[]{3}\cdot5 \\ Long_{\text{ }}length=5\cdot\sqrt[]{3}\approx8.66 \end{gathered}[/tex]