The bank account balance, in dollars, is modeled by the equation f(t) = 1,000. We want to find out about how many years it will take for the account balance to double. We can solve this problem by using the formula for compound interest.
Here is the step-by-step solution:Given, the equation for the bank account balance:f(t) = 1,000To find when the account balance will double, we need to find t such that f(t) = 2,000 (double of 1,000).
That is, we need to solve the following equation for t:1,000 * (1 + r/100)^t
= 2,000
Where r is the interest rate (unknown) and t is the time (unknown).
Divide both sides of the equation by 1,000:(1 + r/100)^t = 2/1= 2
Take the logarithm of both sides of the equation:ln[(1 + r/100)^t] = ln 2Using the property of logarithms, we can bring the exponent t to the front:
tlnt(1 + r/100) = ln 2
Using the division property of logarithms, we can move lnt to the right side of the equation:t = ln 2 / ln(1 + r/100)
We can use the approximation ln(1 + x) ≈ x for small x.
Here x = r/100, which is the interest rate in decimal form. Since r is typically between 1 and 20, we can use the approximation for small values of r/100.
Hence:ln(1 + r/100) ≈ r/100For example,
when r = 10, r/100
= 0.1 and
ln(1.1) ≈ 0.1.
This approximation becomes more accurate as r/100 becomes smaller.Using this approximation,
we get:t ≈ ln 2 / (r/100)
= 100 ln 2 / r
Plug in r = 10 to check the formula
:t ≈ 100 ln 2 / 10
≈ 69.3 years
Therefore, about 69 years (rounded to the nearest year) will be needed for the account balance to double.
Answer: It will take approximately 69 years for the account balance to double.
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Help with these two questions please !!
Answer:
1. (x + 8)(x + 3)
2. (x - 9)²
Step-by-step explanation:
Given trinomials:
1. [tex]x^2+11x+24[/tex]
2. [tex]x^2-18x+81[/tex]
..................................................................................................................................................
When factoring trinomials of the form ax² + bx + c:
Multiply the leading coefficient and the last term.Find the product factors that add up to give you the coefficient of the middle term.Rewrite the polynomial with those factors replacing the middle term...................................................................................................................................................
1) x² + 11x + 24
Step 1: Multiply the leading coefficient (1) and the last term (24).
[tex]\implies 1 \times 24 = \boxed{24}[/tex]
Step 2: Find the product factors that sum up to the middle term's coefficient (11).
[tex]\begin{array}{| c | c |}\cline{1-2} \sf Factors\:of\:24 & \sf Sum\:of\:factors\\\cline{1-2} 1, 24\ \| -1, -24 & 25\ \| -25 \\\cline{1-2} 2, 12\ \| -2, -12 & 14\ \| -14 \\ \cline{1-2} 3, 8\ \| -3, -8 & 11\ \| -11 \\\cline{1-2} 4, 6 \ \| -4, -6 & 10\ \| -10 \\\cline{1-2}\end{array}[/tex]
[tex]\implies 3x+8x=11x[/tex]
Step 3: Rewrite the polynomial with those factors, replacing the middle term.
[tex]\implies x^2+3x+8x+24[/tex]
Step 4: Factor by grouping.
[tex]\implies \overbrace{(x^2+3x)}^x+\overbrace{(8x+24)}^8\ \ \textsf{[ Factor out $x$ and $8$. ]}[/tex]
[tex]\implies x\overbrace{(x+3)}^{\textsf {Factor out}}+8(x+3)[/tex]
[tex]\implies \boxed{(x+8)(x+3)}[/tex]
The factored form of the given trinomial is [tex](x+8)(x+3)[/tex].
..................................................................................................................................................
2) x² - 18x + 81
Step 1: Multiply the leading coefficient (1) and the last term (81).
[tex]\implies 1 \times 81 = \boxed{81}[/tex]
Step 2: Find the product factors that sum up to the middle term's coefficient (-18).
[tex]\begin{array}{| c | c |}\cline{1-2} \sf Factors\:of\:81 & \sf Sum\:of\:factors\\\cline{1-2} 1, 81\ \| -1, -81 & 81\ \| -81 \\\cline{1-2} 3, 27\ \| -3, -27 & 30\ \| -30 \\ \cline{1-2} 9, 9\ \| -9, -9 & 18\ \| -18 \\\cline{1-2}\end{array}[/tex]
[tex]\implies (-9x)+(-9x)=-18x[/tex]
Step 3: Rewrite the polynomial with those factors, replacing the middle term.
[tex]\implies x^2-9x-9x+81[/tex]
Step 4: Factor by grouping.
[tex]\implies \overbrace{(x^2-9x)}^x+\overbrace{(-9x+81)}^{-9}\ \ \textsf{[ Factor out $x$ and $-9$. ]}[/tex]
[tex]\implies x\overbrace{(x-9)}^{\textsf {Factor out}}-9(x-9)[/tex]
[tex]\implies (x-9)(x-9)[/tex]
[tex]\implies \boxed{(x-9)^2}[/tex]
The factored form of the given trinomial is [tex](x-9)(x-9)\ \textsf{or}\ (x-9)^2[/tex].
a sheep started a journey at 6: 30pm on Monday 6th January and finished 68 hours later find the time, date and the day when the journey finished
The time, date and the day when the journey finished is 2:30 pm on Thursday 9th January
TimeTime journey started = 6: 30pm on Monday 6th JanuaryTime taken for journey = 68 hours24 hours = 1 day68 hours = 68/24
= 2 days and 20 hours
6;30 pm on Monday 6th January + 2 days and 20 hours
= 2:30 pm on Thursday 9th January
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HELPPPPPPPPPPPP (-12)^3:8^3
Answer:
-1728:512
Step-by-step explanation:
(-12)³ = -1728
8³ = 512
ratio is -1728:512
Suppose you roll a 6-sided die and draw a card from a deck
of 52 cards. how many possible outcomes are there?
Total possible outcomes are 312
We need to find that if you roll a 6-sided die and draw a card from a deck of 52 cards,what are total possible outcomes
Probability is a measure of the possibility that an event will occur. Many events cannot be predicted with absolute certainty. We can only predict the probability that an event will occur, i.e. how likely it is to occur, using it. Probability can range from 0 to 1, where 0 means an event is impossible and 1 means a certain event
An outcome is the result of an event that depends on probability. Probability refers to the likelihood that something will happen, and an event can have more than one possible outcome.
Enumeration of possible outcomes is the process of determining the number of possible outcomes for an event. There are various methods to carry out this process. The best way to calculate the number of possible outcomes of an event depends on the type of event and the structure of the event
If a 6-sided die and draw a card from a deck,
then each side has total possible outcome 52
so, total number of possible outcomes for 6 - sided dice are 52 * 6
= 312
Hence total number of possible outcomes are 312
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The surface of a table to be built will be in the shape shown below. The distance from the center of the shape to the center of each side is 10.4 inches and the length of each side is 12 inches.
The are of the triangle is mathematically given as
a=62.4in^2. This is further explained below.
What is the area of the triangle?Generally, the equation for the area is mathematically given as
a= 0.5(base X height)
Therefore
a= 0.5 (10.4 x 12)
a=62.4in^2
In conclusion, The area of the triangle is
a=62.4in^2
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HURRY PLEASE I NEED HELP
the number of hours you study for an exam affects your grade.what is the reasonable value of the range?
A - 75
B - 1.25
C - 1/2
D - 30
The reasonable value for the range of the given function is: A. 75.
What of the Range of a Function?The range of a function is the output that a corresponding input value would give as a relation.
In this scenario, the grade you get is a function of number of hours you study. Range is the grade (output), while the domain is hours of study (input).
The given possible range values in the options are, 75, 1.25, -30, 1/2.
A reasonable value of a a grade cannot be a negative number, neither can it be 1.25, nor 1/2.
Therefore, the reasonable value of the range is: A. 75.
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The KM values for the reaction of chymotrypsin with 2 substrates are given below. Which substrate is likely to give a higher value for Vmax
N-acetyltyrosine ethyl ester may likely have a higher Vmax.
How are the Km and Vmax of an enzyme and substrate related?Km is inversely proportional enzyme affinity for substrate. The lower the Km, the higher the enzyme affinity for substrate and vice versa.
Vmax is the maximum velocity of the reaction when the enzyme is fully saturated with substrate.
Based on the Km values, chymotrypsin has higher affinity for N-acetyltyrosine ethyl ester, thus, N-acetyltyrosine ethyl ester may likely have a higher Vmax.
In conclusion, the Vmax is the maximum velocity of an enzyme-catalyzed reaction.
Note that the complete question is given below:
The Km for the reaction of chymotrypsin with N-acetylvaline ethyl ester is 8.8 x 10-2 M, and the Km for the reaction of chymotrypsin with N-acetyltyrosine ethyl ester is 6.6 x 10-4 M.
Which of the following substrates is likely to give a higher value for Vmax?
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Time sensitive!
Find the sum of the first 30 terms of an arithmetic series whose first term is 3 and 30th term is 101. Type answer as an integer (no decimals).
The sum of the first 30 terms of the arithmetic series is 1560
Sum of an Arithmetic progressionFrom the question, we are to determine the sum of the first 30 terms of the series
Using formula,
[tex]S_{n} = \frac{n}{2}(a + l)[/tex]
Where [tex]S_{n}[/tex] is the sum of the nth term
n is the number of terms
a is the first term
and [tex]l[/tex] is the last term
From the given information,
n = 30
a = 3
[tex]l[/tex] = 101
Thus,
[tex]S_{30} = \frac{30}{2}(3+ 101)[/tex]
[tex]S_{30} = 15(104)[/tex]
[tex]S_{30} = 15 \times 104[/tex]
[tex]S_{30} = 1560[/tex]
Hence, the sum of the first 30 terms of the arithmetic series is 1560
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A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be
The null hypothesis be is =15 and the alternative hypothesis be is ≠15
According to the statement
Mean length of time on death row is 17.4 years
Standard deviation of 6.3 years.
Now we determine the hypothesis with mean time 15 years then
P value outcome is 0.00015
So, The null hypothesis be is =15 and the alternative hypothesis be is ≠15
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mann
On a map, 2.5 inches represents 52 miles. What is the actual distance represented by 5 inches on the map?
2.5 5
52 x
Answer:
104 miles.
Step-by-step explanation:
5 inches is 2 * 2.5 inches so the distance required is 2*52 = 104 miles.
Activity in this activity, you will first find the squares of positive and negative numbers. then you will determine whether the solution to an equation is a perfect square. question 1 part a complete the table by squaring each positive x-value listed.
The complete table is shown below.
What is square root?A number's square root is the factor that can be multiplied by itself to yield that number. The square root of, the end square root is the symbol for the square root. Finding an integer's square root is the inverse of squaring a number.Complete the table as follows:
[tex]2^{2} = 2*2 = 4\\3^{2} = 3*3 = 9\\4^{2} = 4*4 = 16\\5^{2} = 5*5=25\\6^{2} =6*6=36\\7^{2} =7*7=49\\8^{2} =8*8=64\\9^{2} =9*9=81\\10^{2} =10*10=100\\11^{2} =11*11=121[/tex]
Therefore, the complete table has been shown.
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The question you are looking for is here:
Complete the table by squaring each positive x-value listed 2, 3,4,5,6,7,8,9,10,11.
What is the solution to this equation? (1/4)^x+1 =31
A. -7/2
B. 2
C. 3/2
D. -2
30 points please help (a) Find an angle between 0 and 2π that is coterminal with −π2.
(b) Find an angle between 0° and 360° that is coterminal with 480°.
The coterminal angles are given as follows:
a) 0º.
b) 120º.
What is the coterminal angle of a negative angle?The coterminal angle of a negative angle between 0 and -360º = -2pi is found adding 360º = 2pi to the angle, hence:
-2pi + 2pi = 0, which is the answer to item a.
What is the coterminal angle of an angle greater than 360º?The coterminal angle is found taking the remainder of the division of the angle by 360º.
The remainder of the division of 480º by 360º is of 120º, hence the coterminal angle of 480º is of 120º.
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Someone help pls, its urgent! ASAP!!! (Geometry)
“Fill in the blanks”
14) theorem
15) sides
16) separately
17) equilateral
18) supplementary
Which values of x would make a polynomial equal to zero if the factors of the
polynomial were (x-5) and (x-6)?
A. -5 and 6
B. 5 and 6
C. 5 and -6
D. -5 and -6
If the factors of the polynomial were (x-5) and (x-6), the values of x that would make the polynomial equal to zero would be 5 and 6.
5 - 5 = 0
6 - 6 = 0
Assuming the polynomial is (x-5)(x-6), if one factor is 0 then anything multiplied by 0 is still 0.
Hope this helps!
Carla left the bookstore traveling 12 mph. Then, 3 hours later, Zari left traveling the same direction at 18 mph. How long does Zari have to travel before catching up with Carla
Answer:
6 hours.
Step-by-step explanation:
Speed = distance / time.
Carla travels 3 * 12 = 36 miles before Zari leaves the shop.
So Zari travels x miles in the same time as Carla travels x- 36 miles.
So 18 = x/t and 12 = (x - 36)/t
From first equation x = 18t so we have the equation
12 = (18t - 36/t
18t - 36 = 12t
6t = 36
t = 6 hours.
Ariel wants to write 8 entries for her blog. After 2 hours,she has 5 entries left. After 4 hours, she has 2 entries left. Which graph represents ariels remaining entries?
The triangles are similar. Find the value of the variable.
Members of a softball team raised $1353 to go to a tournament. They rented a bus for
$943.50 and budgeted $31.50 per player for meals. Write and solve an equation
which can be used to determine x, the number of players the team can bring to the
tournament.
Equation:
Answer: x =
Submit Answer
attempt 1 out of 2
The equation which can be used to determine x, the number of players the team can bring to the
tournament is 1353 = 943.50 + 31.50x and x = 13 players
EquationTotal amount raised = $1353Cost of renting a bus = $943.50Cost of each player's meal = $31.50Number of players in the team = x1353 = 943.50 + 31.50x
1353 - 943.50 = 31.50x
409.50 = 31.50x
x = 409.50 / 31.50
x = 13 players
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An analyst obtains the salaries of all 200 federal appeals court judges in 2017 and computes the mean salary to be $215,000. the $215,000 is a
An analyst obtains the salaries of all 200 federal appeals court judges in 2017 and computes the mean salary to be $215,000. the $215,000 is a parameter.
What is a parameter?A parameter is used to describe the entire population under consideration. For example, we may be interested in the average length of a butterfly. This is a parameter since it contains information about the entire butterfly population.An arbitrary constant whose value identifies a system member (such as a family of curves) is a measure that describes a statistical population (such as a mean or variance).A named variable supplied into a function is referred to as a parameter. Arguments are imported into functions using parameter variables.Therefore, an analyst obtains the salaries of all 200 federal appeals court judges in 2017 and computes the mean salary to be $215,000. the $215,000 is a parameter.
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A lock is opened using a sequence of three numbers. the numbers range from 0 to 39, inclusive, and cannot be repeated within the sequence. what is the probability that all the numbers in the sequence are even? express your answer as a percent and round to the nearest whole number.
Answer:
10%.
Step-by-step explanation:
There are a total of 40 numbers.
Number of permutations of 3 numbers from 40
= 40 P 3
= 40! / (40-3)!
= 59,280.
The total of even numbers is (38 - 2) / 2 + 1 = 19.
Number of permutations of 3
= 19P 3
= 5814
So Prob(all even) = 5814/59280
= 0.098077
= 9.8%
= 10% to nearest whole number.
Find the lateral area of this pyramid whose
base is a regular hexagon with a side length of
3 cm and whose slant height is 12 cm.
12 cm
3 cm
[ ? ] cm2
The lateral area of the given pyramid is 108 cm².
Any three-dimensional figure's lateral surface area is its side surface area.
In the question, we are asked to find the lateral area of the given pyramid, whose base is a regular hexagon, with a side length of 3 cm, and the slant height of the pyramid is 12 cm.
To determine the lateral area, we determine the area of one lateral face, and then multiply it by the number of lateral faces in the pyramid.
The number of lateral faces = sides of the base = 6 {Since the base is a regular hexagon}.
Area of a lateral face = (1/2)*base*height {Since its a triangle},
or, area of a lateral face = (1/2)*3*12 {Since the base is the side length of the base, and the height is the slant height},
or, area of a lateral face = 18 cm².
Thus, the lateral area of the pyramid = 6 * 18 cm² = 108 cm².
Thus, the lateral area of the given pyramid is 108 cm².
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How did the temperature change if :
At first, it increased by 5% and then increased by 20%
So my question is,
What would the process be for solving this (Please answer it in a way a middle schooler can understand!)
===================================================
Explanation:
If it increases by 5%, then we use the multiplier 1.05
Think of it like saying 100% + 5% = 1 + 0.05 = 1.05
Similarly, an increase of 20% means we involve 1.20
Combine those multipliers to get: 1.05*1.20 = 1.26
This is a net increase of 26%
Interestingly, we get fairly close to 25% which is what many students might go for (since 5 + 20 = 25). This is a beginner's trap.
----------------
An example:
Let's say the temperature is 20 degrees Celsius (68 degrees Fahrenheit)
An increase of 5% means it moves to 1.05*20 = 21 degrees C
A 20% increase means we go to 1.20*21 = 25.2 degrees C
The shortcut would be to do a 26% increase to get 1.26*20 = 25.2 to help confirm we have the correct combined increase.
A simple main effect analysis examines: relationships between independent and dependent variables. the overall effect of the interaction. the overall effect of one independent variable. mean differences at each level of the independent variable.
A simple main effect analysis looks at the mean difference at each level of the independent variable.
Given With a simple main effect, the analysis examines.
Simple effects are also called the main effects of design of experiments. The main effect can be defined as the effect of the explanatory variables measured on a particular criterion or response variable of the study. It's easy to say that what happens within the level of the explanatory variables is the difference. In a factorial experiment, the simple main effect is that one independent variable is at a specified level of another. The main effect of one independent variable is averaged across the levels of the other independent variable. For a simple main effect, the results are analyzed as if a separate experiment was performed at each level of the other independent variables.
Therefore, a simple main effect analysis looks at the mean difference at each level of the independent variable.
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Help, I don’t know what I’m doing
Trigonometry
Answer:
[tex]\frac{\sqrt{3} }{6}[/tex]
Step-by-step explanation:
using the 30- 60- 90 triangle for exact values , then
tan60° = [tex]\sqrt{3}[/tex] , and
cot60° = [tex]\frac{1}{tan60}[/tex] = [tex]\frac{1}{\sqrt{3} }[/tex]
cos60° = [tex]\frac{1}{2}[/tex]
cot60° cos60°
= [tex]\frac{1}{\sqrt{3} }[/tex] × [tex]\frac{1}{2}[/tex]
= [tex]\frac{1}{2\sqrt{3} }[/tex] ← rationalise the denominator by multiplying by [tex]\frac{\sqrt{3} }{\sqrt{3} }[/tex]
= [tex]\frac{1}{2\sqrt{3} }[/tex] × [tex]\frac{\sqrt{3} }{\sqrt{3} }[/tex]
= [tex]\frac{\sqrt{3} }{6}[/tex]
The answer is [tex]\boxed {\frac{1}{2\sqrt{3}}}[/tex]
The needed trigonometric values :
The value of cot 60° = 1/√3The value of cos 60° = 1/2Hence, the value of the expression is :
cot 60° × cos 60°1/√3 × 1/21/2√3in quadrilateral ABCD, AB || CD. which additional piece of information is needed to determine the ABCD is a parallelogram?
The additional information needed to determine that ABCD is a parallelogram is: A. AB ≅ CD.
How to Identify a Parallelogram?One of the criteria for proving that a quadrilateral is a parallelogram is that one of the pairs of opposite sides are congruent and also parallel to each.
We are already given that side AB is parallel to side CD, therefore, the additional information we would need to prove that quadrilateral ABCD is a parallelogram is: A. AB ≅ CD.
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Solve:
3⁄2 c + 23 = 38
c = ___
Answer:
(C)=22.5
Step-by-step explanation:
Hi! ❄
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
First, let's subtract by 23 both sides.
[tex]\sf{\dfrac{3}{2}c=15}[/tex]
Now we multiply the equation by 2.
[tex]\sf{3c=30}[/tex]. See how on the left hand side, the 3/2c*2 turned into 3c? That's because multiplication and division are operations that undo each other.
And Now we just divide by 3 both sides.
[tex]\sf{c=10}[/tex]
Hope this made sense !!
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
[tex]\star\tiny\pmb{calligraphy}\star[/tex]
Find the product -2/9(-5/3) I-Ready math
The product of the given fractions, -2/9(-5/3), is: 10/27.
How to Find the Product of Two Fractions?Given the fractions, -2/9(-5/3), recall that minus multiplied by minus equals plus.
So therefore, the product of the two fractions would be a positive number. Thus:
-2/9(-5/3) = (-2 × -5)/(9 × 3)
-2/9(-5/3) = 10/27
Therefore, the product is: 10/27.
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What do you do in calculus?
Answer:
In math, calculus deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two basic types of calculus are differential calculus and integral calculus.
Step-by-step explanation:
hope it helps
sorry if I'm wrong
Please help me, random answers will be reported
Answer:
the side length of hypotenuse is 16.28
Step-by-step explanation:
[tex]a^{2}+b^{2}=c^{2}[/tex]
= [tex]11^{2}+12^{2}= C^{2}[/tex]
= [tex]121+144=c^{2}[/tex]
= [tex]265= c^{2}[/tex]
= [tex]\sqrt{265} = c[/tex]
= [tex]c = 16.28\\[/tex]
Answer:
A = 42.5° (nearest tenth)
B = 47.5° (nearest tenth)
C = 90°
a = 11
b = 12
c = √(265) = 16.3 (nearest tenth)
Step-by-step explanation:
Information:
Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.
From inspection of the triangle:
a = 11b = 12C = 90°To find the missing side length, use Pythagoras Theorem:
Pythagoras Theorem
a² + b² = c²
(where a and b are the legs and c is the hypotenuse of a right triangle).
Substitute the given values of a and b into the formula and solve for c:
⇒ 11² + 12² = c²
⇒ 121 + 144 = c²
⇒ c² = 265
⇒ c = √(265)
To find one of the missing angles, use the Sine Rule.
Sine Rule
[tex]\sf \dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}[/tex]
(where A, B and C are the angles and a, b and c are the sides opposite the angles)
Substitute the given values into the formula:
[tex]\implies \sf \dfrac{\sin A}{11}=\dfrac{\sin B}{12}=\dfrac{\sin 90^{\circ}}{\sqrt{265}}[/tex]
Solve for angle A:
[tex]\implies \sf \dfrac{\sin A}{11}=\dfrac{\sin 90^{\circ}}{\sqrt{265}}[/tex]
[tex]\implies \sf \dfrac{\sin A}{11}=\dfrac{1}{\sqrt{265}}[/tex]
[tex]\implies \sf \sin A=\dfrac{11}{\sqrt{265}}[/tex]
[tex]\implies \sf A=\sin^{-1} \left(\dfrac{11}{\sqrt{265}}\right)[/tex]
[tex]\implies \sf A=42.51044708...^{\circ}[/tex]
Therefore, A = 42.5° (nearest tenth)
Interior angles of a triangle sum to 180°:
⇒ A + B + C = 180°
⇒ 42.5° + B + 90° = 180°
⇒ B + 132.5° = 180°
⇒ B = 47.5°
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