Factorization of sin²θ + sin²θ cos²θ gives us; 2sin²θ - sin⁴θ
How to simplify Trigonometric Functions?
We want to simplify;
sin²θ + sin²θ cos²θ
Let us factorize out sin²θ to get;
sin²θ(1 + cos²θ)
Now, we know that cos²θ = 1 - sin²θ
Thus;
sin²θ(1 + cos²θ) = sin²θ(1 + 1 - sin²θ)
⇒ sin²θ(2 - sin²θ)
⇒ 2sin²θ - sin⁴θ
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Using the information in the diagram, what is the height of the tree to the nearest foot?
45 ft
54 ft
60 ft
135 ft
Answer:
60 ft
Step-by-step explanation:
a = 90
b = 108ft
the base angle can be found by using sin(x) = 120/108x2
=> x = 33.7 degrees
To find height, use tan(33.7) = Opp/ Adjacent = x / 90
=> x = 60.02 or 60 ft
A rectangle has sides of length 8a cm and 7a cm respectively. The perimeter of the rectangle is 42 cm more than the perimeter of a square with side 4a cm. Find the length of the side of the square.
Answer:
Length of the side of the square is 12 cm.
Step-by-step explanation:
Given information:
[tex]\text{side}_1 = 8a \text{ cm (side of a rectangle)}\\\text{side}_2 = 7a \text{ cm (side of a rectangle)}\\P_\text{rec} = 42 + P_\text{sq} \text{ cm } (P_\text{rec} \text{ is perimeter of the rectangle and } \\P_\text{sq} \text{ is the perimeter of the square)}\\\text{side}_\text{sq} = 4a \text{ cm (side of a square)}[/tex]
Our mission is to find the length of the side of the square. We know that the side of the square is 4a cm, so to be able to answer we have to find the value of a first.
Step 1: Finding the value of a
Key to finding the value of a is the perimeter of the rectangle. Notice that we can calculate the perimeter of the rectangle in two different ways.
One way was given in the question:
[tex]P_\text{rec} = 42 + P_\text{sq}[/tex]
We want equations in terms of a, so we can calculate a. Perimeter of the square can be calculated as:
[tex]\text{perimeter}_\text{square} = 4 \times \text{side}[/tex]
Substituting the values in equation:
[tex]P_\text{sq} = 4 \times 4a\\P_\text{sq} = 16a[/tex]
Therefore the first equation for perimeter of the rectangle becomes:
[tex]P_\text{rec} = 42 + P_\text{sq}\\\fbox{\begin{test}P_\text{rec} = 42 + 16a\end{test}}[/tex]
For the second way we use formula for the perimeter of rectangle which is:
[tex]\text{perimeter}_\text{rectangle} = 2 \times \text{width} + 2 \times \text{length}[/tex]
Substituting the values in equation:
[tex]P_\text{rec} = 2 \times 7a + 2 \times 8a\\P_\text{rec} = 14a + 16a\\\fbox{\begin{test}P_\text{rec} = 30a\end{test}}[/tex]
Now let's equate both perimeters of the rectangle.
[tex]P_\text{rec} = P_\text{rec}\\42+16a = 30a\\42 =14a\\3 = a[/tex]
Step 2: Find the length of the side of the square
It's given that side of the square is 4a cm. All we have to do is substituting a with its value, which we got in the previous step.
[tex]\text{side}_\text{sq} = 4a\\\text{side}_\text{sq} = 4(3)\\\text{side}_\text{sq} = 12 \text{ cm}[/tex]
Frank has $500 in a savings account that earns 2% interest per year. The interest is not compounded. How much interest will Frank earn in 5 years?
Answer:
$50 interest
Step-by-step explanation:
Simple interest formula
I = Prt
where:
I = interested earnedP = principal amountr = interest rate (in decimal form)t = time (in years)Given:
P = $500r = 2% = 0.02t = 5 yearsSubstitute the given values into the formula and solve for I:
⇒ I = 500 · 0.02 · 5
⇒ I = 50
Therefore, Frank will earn $50 interest in 5 years.
Graph the function f(x) = 2^x - 7 on the set of axes, if g(x) = 1.5x - 3 determine if f(x) > g(x) when x - 4
The inequality expression f(4) > g(4) is true when x = 4
How to determine if f(x) > g(x)?The functions are given as:
[tex]f(x) = 2^x - 7[/tex]
g(x) = 1.5x - 3
See attachment for the graph of both functions.
From the graph, we have:
f(4) = 9
g(4) = 3
This means that:
f(4) > g(4) when x = 4
Hence, the inequality f(4) > g(4) is true when x = 4
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Which of the triangles shown are scalene triangles? (2 points)
A group of triangles. Triangle A has three acute angles and two equal sides. Triangle B has two acute angles, one ninety degree angle, and two equal sides. Triangle C has three acute angles and two equal sides. Triangle D has two acute angles and one obtuse angle. All the sides on this triangle are different lengths.
Triangle A and Triangle B
Triangle C
Triangle B and Triangle D
Triangle D
Answer:
Triangle D.
Step-by-step explanation:
A scalene triangle has different sides.
Let x and y be directly related such that y = 9 when x = 3. Find y when x = 7.
Answer:
y=21
Step-by-step explanation:
y= 9 cos x =3 so y=3 when x=1 on the base level so if x=7 would be x=1 on the base level * 7 then y=3 on the base level * 7 would be y = 21
Which of the following is true about a function that is constant?
It is a horizontal line.
It moves in an upward direction.
It moves in a downward direction.
It is a curve.
A function assigns the values. The correct option that describes a constant function is that It is a horizontal line.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
The correct option that describes a constant function is that It is a horizontal line.
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Answer: the answer is A the others are linear or higher
It moves in a horizontal line. Its 'y value' does not change.
Step-by-step explanation:
Which of the following graphs shows the solution for the inequality
y-4> 2(x+2)?
Answer:
Answer:
Graph C shows the solution for the inequality
Step-by-step explanation:
y - 4 > 2(x + 2)
y > 2x + 4 + 4
y > 2x + 8
Let y = 2x + 8, Then
X-intercept (0, 8)
Y-intercept (-4, 0)
Since the inequality sign is > we use broken line.
Put, (0, 0)
y > 2x + 8
0 > 0+ 8
0 > 8 Which is false
so, the answer will be above the broken line
Hence , Graph C shows inequality
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Which equations represent inverse variation? Check all that apply.
y=2x
pv = 13
z = ²/x
4=v/x
h=9g/5
The equations representing inverse variation are: pv = 13 and z = (2 / x) , option B and C are the answer.
What is an Inverse Variation ?When on increasing one quantity the other quantity decreases , such variation between the two entities is called Inverse Variation
For this case, what you should know is that the equations that represent an inverse variation are those that could not form a straight line, for example.
The equations are given in the question and it has to be checked if they represent inverse variation
y = 2x
y and x are in a direct variation
pv = 13
p = 13 / v
P and v are represent an inverse variation.
z = (2 / x)
z and x are represent an inverse variation.
4 = v/x
v =4x
v and x are in direct variation
h=9g/5
h and g are in direct variation
therefore , the equations representing inverse variation are:
pv = 13 and z = (2 / x)
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For what values of t can 10x2+tx+8 be written as the product of two binomials?
Answer:
7
Step-by-step explanation:
The age of a father is 2 less than 7 times the age of his son. In 3 years, the sum of their ages will be 52. If the son’s present age is s years, which equation models this situation? A. B. C. D.
Answer:
First,
the age of the son: s
the age of the father is 2 less than 7 times the age of his son: 7s - 2
In 3 years, the sum of their ages will be 52, In 3 years, the age of the sun will
increase 3, so s+3. Also, the age of his
father also increase 3, (7s-2)+3
(s+3)+ [(7s-2)+3] = 52 is the equation.
The correct answer is :
( 7s - 2 ) +3+ ( s + 3 ) = 52 , or 8s + 4 = 52 .
Explanation :Since s is the son's age , " two less than seven times " the son's age would be represented by 7s - 2 .
To represent this in 3 years , we would add 3 : ( 7s - 2 ) +3 .
In 3 years , the son's age , s , would be represented by s + 3 . We are told that the sum of these ages will be 52 ; this gives us ( 7s - 2 ) +3+ ( s + 3 ) = 52 .
To simplify this , combine like terms .
7s + s = 8s ; -2 + 3 + 3 = 4 .
This gives us 8s + 4 = 52 .
Slope of (3,2) and (19,6)
Step-by-step explanation:
[tex]slope = \frac{y2 - y1}{x2 - x1} \\ = \frac{6 - 2}{19 - 3} \\ = \frac{4}{16} \\ \boxed{slope = \frac{1}{4} }[/tex]
Only answer if you knoww
The number of 15-year old students that took part in the survey is 300 students and 20 "15-year old" students spent between 35 and 45 minutes on social media.
What is a frequency polygon?A frequency polygon can be defined as a type of line graph that is used to plot a class frequency against the class midpoint.
For the 15-year old students, the total number of students would be calculated by adding all the data points at each frequency:
Total students = 30 + 30 + 50 + 60 + 40 + 50 + 40
Total students = 300 students.
Part B.In order to determine the number of 15-year old students than 11-year old that spent between 35 and 45 minutes on social media, we would calculate the average and then read it from the frequency polygon.
Average = (35 + 45)/2
Average = 80/2
Average = 40.
For the 11-year old, we have:
Frequency (40) = 40
For the 15-year old, we have:
Frequency (40) = 60
Now, we can calculate the difference as follows:
Difference = 60 - 20
Difference = 20 students.
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I’m having trouble converting mixed fractions to proper fractions can someone explain?
Answer:
See below
Step-by-step explanation:
To get the fraction from mixed number you just need to take the number standing alone, multiply with the denominator and add with the numerator and keep it on top with the denominator in place to turn it into a fraction.
For example, [2]3/4 = 2*4 + 3 = 11 => the fraction is 11/4.
You translate a line 10 units up. Is the image of the line still a line? O Yes O No
Answer:
It depends on what way the line is facing, if it starts out vertical(meaning up or down), the answer is yes.
Please help me answer this question.
Answer:
86%
Step-by-step explanation:
each shaped area represent 1% so you divide or subtract the unshaped area with the shaped area.
Use the function below to find F(3)
F(x) = (1/6)^x
Answer: 1/216
Step-by-step explanation: please mark me brainly
Answer:
f(3)=0.005
Step-by-step explanation:
Plug in 3 for x
f(3)=(1/6)^3
f(3)=0.005
Hope this helps!
If not, I am sorry.
△ABC ~ △AMN and AM = 6, MB = 4, AN = 8, then what is the value of AC?
Answer:
AC = 13.3
Make proportional relationship:
[tex]\sf \dfrac{AB}{AC} = \dfrac{AM}{AN}[/tex]
Insert values
[tex]\rightarrow \sf \dfrac{10}{AC} = \dfrac{6}{8}[/tex]
Cross multiply
[tex]\rightarrow \sf AC = \dfrac{10(8)}{6}[/tex]
Simplify
[tex]\rightarrow \sf AC =13.3[/tex]
In her monthly tests, Mary got 20 more marks in Mathematics than in English. The total of her
marks for both tests were 130. Find her marks in each test
Answer:
75 on mathematics and 55 on English!
Step-by-step explanation:
Answer:
English test score: 55
Mathematics test score: 75
Step-by-step explanation:
Let [tex]e[/tex] be Mary's score on the English test.
We can say that [tex]e + 20[/tex] is her score on the Mathematics test, as the score for the Mathematics test is just the English test score but 20 higher.
Since both scores add up to 130, we can set up an equation to solve for the English test score. Since we said [tex]e + 20[/tex] is Mary's score on the Mathematics test, [tex]e + e + 20[/tex] is the total of her 2 test scores.
[tex]e + e + 20 = 130\\2e + 20 = 130\\2e = 110\\\boxed{e = 55}[/tex]
This means that Mary got a 55 on her English test score.
To solve for the Mathematics test score we can substitute 55 for [tex]e[/tex] in [tex]e + 20[/tex].
[tex]e + 20 = 55 + 20 \rightarrow \boxed{75}[/tex]
This means Mary got a 75 on her Mathematics test.
Those are the answers
- Kan Academy Advance
How many fourths are in 3
4
?
So, there are 3 fourths in 3 fourths
Ginger has 52 stamps and 68 regular
stickers. What is the ratio of stamps to
regular stickers? What is the simplified
ratio?
Answer:
13 stamps : 17 stickers
Step-by-step explanation:
52 stamps : 68 stickers can be simplified down to 13 stamps : 17 stickers because their greatest common factor is 4
4 there are only red sweets and yellow sweets in a bag.
there are n red sweets in the bag.
there are 8 yellow sweets in the bag.
sajid is going to take at random a sweet from the bag and eat it.
7
he says that the probability that the sweet will be red is
10
7
(a) show why the probability cannot be
10
after sajid has taken the first sweet from the bag and eaten it, he is going to take at
random a second sweet from the bag.
given that the probability that both the sweets he takes will be red is
uw
(b) work out the number of red sweets in the bag.
you must show all your working.
Using the probability concept, it is found that since the number of red sweets would be a decimal number, the probability cannot be [tex]\dfrac{7}{10}[/tex]
What is probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
In this problem:
In total, there are 8 + n sweets in the bag.Of those, n are red.The probability of red is:
[tex]p=\dfrac{n}{n+8}[/tex]
Supposing, we solve for n:
[tex]\dfrac{n}{n+8} = \dfrac{7}{10}[/tex]
10n = 7n + 56
3n = 56
n = 56 / 3
n = 18.67
Since the number of red sweets would be a decimal number, the probability cannot be 7 / 10
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The sum of the volume of two rectangular prisms, box a and box b are 14.325cm cubed. Box a has a volume of 5.61cm.
The equation that could be used to determine the volume of Box B is (B = 14.325cm³ - 5.61cm³) and the volume of Box B is 8.715cm³.
This question is incomplete, the complete question is:
The sum of the volume of two rectangular prisms, box a and box b are 14.325cm cubed. Box a has a volume of 5.61cm.
a. Let B represent the volume of Box B in cubic centimeters. Write an equation that could be used to determine the volume of Box B.
b. Solve the equation to determine the volume of Box B
What is volume?Volume is simply the amount of space that is enclosed within a container.
Given that;
Sum of the volume of two rectangular prisms A&B = 14.325cm³Volume of Box A = 5.61cm³a)
Let B represent the volume of Box B in cubic centimeters. The equation that could be used to determine the volume of Box B will be;
Sum of the volume of two rectangular prisms A&B = Volume of Box A + Volume of Box B
14.325cm³ = 5.61cm³ + B
B = 14.325cm³ - 5.61cm³
Hence, The equation that could be used to determine the volume of Box B is B = 14.325cm³ - 5.61cm³
b)
To determine the volume of Box B, we solve the equation.
B = 14.325cm³ - 5.61cm³
B = 8.715cm³
Hence, the volume of Box B is 8.715cm³.
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What is the outlier of 35, 39, 35, 34, 38, 39, 38, 33, 88, 33, 35, 38
Answer:
88
Step-by-step explanation:
All of the numbers are in the thirties (30's) except for 88, so that is the outlier
Concentrate is diluted with water in the ratio 3 : 2. How much concentrate is there in 125 mL of total solution?
Answer:
75 mL.
Step-by-step explanation:
3/(3+2) = 3/5 of the solution is concentrate.
So the answer is 125 * 3/5
= 3*25
= 75 mL.
A gift shop owner bought fifty vases for $210. she sells seven vases for what she paid for ten originally. what profit does she make on the fifty vases, which were all sold?
The profit earned by the gift shop owner on fifty vases will be $ 91.07.
What is profit?The amount of money earned by any seller on the cost price is called the profit. It is given that A gift shop owner bought fifty vases for $210. she sells seven vases for what she paid for ten originally.
The profit will be calculated as we will create the expressions for all the selling and purchasing given in the question.
The total cost price of vases:-
50 vases = $210
1 Vase = 210 / 50 = 4.25
So 10 vases will be purchased at the price of:-
10 Vase = 4.25 x 10 = 42.5
Now from the question of the amount he paid for 10 vases, he gained it by selling 7 vases.
7 Vase = 42.5
1 Vase = 42.5 / 7 = 6.07
So he sold one vase for $6.07 and purchased it at $4.25.
The profit will be = 6.07 - 4.25 = 1.821
So profit for selling 50 vases will be:-
Profit = 50 x 1.821 = $91.07
Therefore profit earned by the gift shop owner on fifty vases will be $ 91.07.
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A conditional relative frequency table is generated by column from a set of data. the conditional relative frequencies of the two categorical variables are then compared. if the relative frequencies being compared are 0. 21 and 0. 79, which conclusion is most likely supported by the data? an association cannot be determined between the categorical variables because the relative frequencies are not similar in value. there is likely an association between the categorical variables because the relative frequencies are not similar in value. an association cannot be determined between the categorical variables because the sum of the relative frequencies is 1. 0. there is likely an association between the categorical variables because the sum of the relative frequencies is 1. 0.
The conclusion is there is likely an association between the categorical variables because the sum of the relative frequencies is 1. 0.
What is the true statement?
Categorical variables are variables that can only take on a fixed number of values. An example of categorical variables is gender. One can either be a male or a female.
The sum of the categorical variables is 1. This indicates that the two variables are negatively correlated. It means that someone would be either pick one of the variables.
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Answer:
c
Step-by-step explanation:
How would you write the equation with a slope of 2/3 and a y-intercept of -3
y = (3/2)x + 3
y = (2/3)x - 3
y = 2/3 + 3
y = 3/2 - 3
Answer:
y = (2/3)x - 3
Step-by-step explanation:
The general structure of an equation in slope-intercept form is:
y = mx + b
In this equation, "m" represents the slope and "b" represents the y-intercept. You were given the slope and y-intercept, so finding the final equation is just a matter of substituting the variables.
slope = m = (2/3)
y-intercept = b = -3
y = mx + b
y = (2/3)x + (-3)
y = (2/3)x - 3
Write the exponent for the expression.
The exponent for the expression is
The exponent for the expression, 5 × 5 × 5 × 5 × 5, would be expressed as: [tex]5^5[/tex].
What is an Exponent?An exponent can be defined as the power to which the base is to b e raised.
For example, a × a × a can be expressed in exponent form as a³.
In the same vein, given the expression, 5 × 5 × 5 × 5 × 5, we would expressed this in exponent form as: [tex]5^5[/tex].
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Factor 125x9 +64.
O (5x³-4)(25x5+20x³
+ 16)
+ 16)
O (5x³-4)(25x3+20x³
O (5x³+4)(25x5 - 20x³ +16)
O (5x³+4)(25x³ - 20x³ + 16) hurry plsss
[tex]~~~~125x^9+64\\\\=(5x^3)^3+4^3\\\\=(5x^3 +4)[(5x^3)^2 - (5x^3)(4) + 4^2]\\\\=(5x^3 +4)(25x^6-20x^3+16)[/tex]