The cost of each strand of Christmas lights is $4, and the cost of each decoration is $3.5 which is calculated using equations 3x + 6y = 33 and 2x + 8y = 36.
Let x= price in dollars of each strand of Christmas lights, and let y =price in dollars of each ornament. Then, we can set up the following system of equations:
3x + 6y = 33
2x + 8y = 36
The first equation represents the amount spent by the first customer, who bought 3 strands of Christmas lights and 6 ornaments for $33. The second equation represents the amount spent by the second customer, who paid $3 more than the first customer for 2 strands of Christmas lights and 8 ornaments.
To solve for x and y, we can use either substitution or elimination. Here, we will use elimination:
Multiplying the first equation by 2, we get:
6x + 12y = 66
Multiplying the second equation by 3, we get:
6x + 24y = 108
Subtracting the first equation from the second equation,
12y = 42
Dividing both sides by 12, we get:
y = 3.5
Substituting y = 3.5 into the first equation, we get:
3x + 6(3.5) = 33
Simplifying, we get:
3x + 21 = 33
Subtracting 21 from both sides, we get:
3x = 12
Dividing both sides by 3, we get:
x = 4
therefore, the cost of each strand of Christmas lights is $4, and the cost of each decoration is $3.5.
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The medal distribution from the 2021 summer Olympic Games is shown in the table what is the probability that the winner won a gold medal and is from the United States?
The probability of randomly selecting a gold medalist from the US is0.019
How to find the probability?First, the probability of winning a gold medal is given by the quotient between the number of gold medals and the total number of medals.
The number of medals is:
gold = 48 + 90 + 107 + 114 + 397 = 756
There are 1,019 silver ones
There are 782 bronze ones
Then the total number is 756 + 1019 + 782 = 2,557
Then the probability for the gold medal is:
P = 756/2,557
And the probability that the gold medalist is from the US is given by the quotient between the number won by the US (48) and the total number of gold ones (756)
So we get.
Q = 48/756
The joint probability is given by the product of the two:
p = (756/2,557 )*(48/756) = 48/2,557 = 0.019
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A skydiver s jumps from a plane at an altitude of 3km the radius of the earth is approximately 6400km how far is the horizon h from the skydiver when she leaves the plane calculate this distance to the nearest kilometer
Answer:
196 km
Step-by-step explanation:
See attached figure
The skydiver is at a height of 3 km above the earth surface at point S
Point H represents the horizon as seen by the skydiver
CH represents the radius of the earth = 6400
SH is the line of sight from S to H and is tangential to the circle representing the earth's surface
Sine SH is tangent to the circle, CH is perpendicular to SH
Therefore the triangle CSH is a right triangle with m∠H = 90°
The hypotenuse of the triangle is CS = 6400 + 3 = 6403 km
The legs are CH = 6400 and SH which is to be determined
Using the Pythagorean theorem
CS² = CH² + SH²
Plugging in knowns:
6403² = 6400² + SH²
SH² = 6403² - 6400² = 38409
SH = √(38409) ≈ 195.98 km ≈ 196 km rounded to nearest km
What is the term used to denote a number raised to the power of three?
The term used to denote a number raised to the power of three is cubed.
When a number is cubed, it is multiplied by itself three times. For example, 2 cubed (written as 2³) is equal to 2 x 2 x 2, which equals 8.
Cubing a number is a common mathematical operation that is used in a variety of applications, including geometry, physics, and engineering. It is also used in basic arithmetic, such as finding the volume of a cube.
Cubing a number is similar to squaring a number, which is when a number is raised to the power of two. However, cubing a number results in a much larger value than squaring the same number. For example, 2 squared (written as 2²) is equal to 2 x 2, which equals 4, while 2 cubed (written as 2³) is equal to 2 x 2 x 2, which equals 8.
In summary, the term used to denote a number raised to the power of three is "cubed". Cubing a number involves multiplying the number by itself three times, and is a common operation in various fields of mathematics and science.
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1. Why is an understanding of Boolean algebra important to computer scientists?
An understanding of Boolean algebra is important to computer scientists because it provides the basis for logical operations that are used in computer programming and digital electronics.
Boolean algebra is a branch of mathematics that deals with binary variables and logic gates. Computer scientists use Boolean algebra to design and analyze digital circuits and logic gates, as well as to create programming languages and algorithms. The ability to understand and apply Boolean algebra is crucial for computer scientists in order to develop efficient and reliable software and hardware systems.
Boolean algebra deals with binary values (true/false or 1/0) and uses logical operations like AND, OR, and NOT to manipulate them. Computer scientists use these concepts in programming, data structures, algorithms, and optimization. By mastering Boolean algebra, computer scientists can create efficient and reliable software and hardware systems.
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Which of the two functions below has the largest maximum y-value?
f(x)=-3x4-14
g(x) = -x³ + 2
Answer:
G. (x) has the largest
Step-by-step explanation:
maximum y value
10) What is SS in an ANOVA summary report?
Statistical score
Square summary
Sum of squares
Summary statistics
In an ANOVA summary report, SS stands for Sum of Squares.
It is a key term used in the analysis of variance (ANOVA) technique, which helps in assessing the differences among group means in a sample. The Sum of Squares is calculated to partition the total variability in the dataset into components attributable to different sources.
The ANOVA summary report typically includes the following elements: Source of variation (between groups, within groups, and total), Degrees of Freedom (DF), Sum of Squares (SS), Mean Squares (MS), F-ratio, and the p-value. The Sum of Squares is essential in determining the Mean Squares, which is calculated by dividing the SS by the corresponding degrees of freedom. The F-ratio is then derived by dividing the Mean Square Between Groups by the Mean Square Within Groups.
In conclusion, the term SS in an ANOVA summary report refers to the Sum of Squares, which represents the total variability in the dataset divided into components based on different sources. This concept is vital for conducting the analysis of variance and assessing the statistical significance of differences among group means.
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Answer this asap 3x²+2(x+2)-3x(2x+1)
Answer: -1 (x-1) (3x+4)
Step-by-step explanation: You have to dristibute, combine Like terms, find the common factors, use the sum- product pattern, common factor from 2 pairs, re-write in correct form then you got your answer
how many bit strings of length 256 have exactly 150 0's while also having no double 1's in the sequence? explain your answer
There are 46 bit strings of
length
256 that have exactly 150 0's and no double 1's in the sequence.
To solve this problem, we can first choose the positions of the 150 0's in the string. This can be done in $\binom{256}{150}$ ways.
Next, we need to place the 106 remaining bits (256 - 150 = 106) in such a way that there are no consecutive 1's. We can approach this problem recursively. Let $f(n)$ be the number of valid bit strings of length $n$. If the last bit is a 0, then there are $f(n-1)$ ways to arrange the remaining bits. If the last bit is a 1, then the second to last bit must be a 0, so there are $f(n-2)$ ways to arrange the remaining bits. Therefore, we have the recurrence relation $f(n) = f(n-1) + f(n-2)$ with
initial conditions
$f(1) = 2$ and $f(2) = 3$ (since we cannot have two consecutive 1's).
Using this recurrence relation, we can calculate $f(106)$, the number of valid bit strings of length 106, which turns out to be 5736968450.
Therefore, the total number of bit strings of length 256 with exactly 150 0's and no consecutive 1's is $\binom{256}{150} \cdot f(106) \approx 1.836 \times 10^{60}$.
Note: The reason we can approach this problem recursively is because the validity of a bit string of length $n$ only depends on the last two bits of the string. Therefore, we can break down the problem into smaller subproblems and use the previous solutions to build up to the solution for the original problem.
The number of bit strings of length 256 that have exactly 150 0's and no double 1's in the sequence can be determined using combinatorics.
First, consider that there will be 150 0's and 106 1's in the
sequence
. As we don't want any consecutive 1's, we can place each 1 between the 0's. Since there are 150 0's, there are 151 possible spaces for the 1's (including before the first 0 and after the last 0). Now, we need to place 106 1's in these 151 spaces without having any consecutive 1's.
We can treat this as a problem of
distributing
106 1's into 151 "bins" (spaces between 0's). To avoid having consecutive 1's, we can place at most 1 bit in each bin. To achieve this, we can first place 105 1's into 105 of the 151 bins. Now, we have 1 bit left to place and 46 bins remaining (151 - 105 = 46).
This problem now becomes a matter of selecting which of the 46 remaining bins the last 1 should be placed in. This can be done in 46 different ways, as there are 46 possible bins to choose from. Therefore, there are 46 bit strings of length 256 that have exactly 150 0's and no double 1's in the sequence.
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Amy made 13 quarts of pink lemonade and 3 gallons of yello lemonade. How many quarts of lemonade did she make?
Can anybody help me please? Thanks
Answer: 16 quarts
Step-by-step explanation:
Just add
5. Given the graph below, determine whether each statement is true or false.
Statement
-3 is a zero of the function
-1 is a zero of the function
1 is a zero of the function
2 is a zero of the function
4 is a zero of the function
True False
There are only two true statements, these are are:
-1 is a zero of the function2 is a zero of the functionWhich statements are true?We want to see which of the given statements is true. Remember that a zero of a function is a value of x such that the function intercepts the x-axis.
Looking at the graph, we can see that the zeros are at:
x = -1
x = 2
Then the true statements are:
-1 is a zero of the function2 is a zero of the functionAll the other ones are false.
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PLS HELP HURRY ILL GIVE BRAINLIEST
The triangular prism has a surface area of 1340 square meters. The answer is 1340 m².
How to calculate surface area?To calculate the surface area of a triangular prism, find the area of all its faces and add them together.
The triangular base has a base of 10 m and a height of 14 m, so its area is:
(1/2) × base × height = (1/2) × 10 m × 14 m = 70 m²
There are two identical triangular faces, so the total area of both is:
2 × 70 m² = 140 m²
The rectangular faces have dimensions of 10 m by 25 m and 14 m by 25 m, so their areas are:
10 m × 25 m = 250 m²
14 m × 25 m = 350 m²
Again, there are two rectangular faces, so the total area of both is:
2 × (250 m² + 350 m²) = 1200 m²
Finally, add the areas of all the faces:
140 m² + 1200 m² = 1340 m²
Therefore, the surface area of the triangular prism is 1340 square meters.
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how can i answer it
The correct statements in the given options are, x = 51, y = 34, and m ∠1 = 44⁰
options, B,D,E
What is the x and y?The value of x and y can be determined by applying the following equation.
3y - 18 = 84 (vertical opposite angles are equal)
3y = 102
y = 102/3
y = 34
The value of x is calculated as follows;
∠1 + 3y - 18 + ²/₃(x + 27) = 180 (sum of angles on a straight line)
∠1 + 3(34) - 18 + ²/₃(x + 27) = 180
∠1 + ²/₃(x + 27) = 96
∠1 = 96 - ²/₃(x + 27) -------(1)
3x - 25 = 3y - 18 + ∠1 ---- (2)
3x - 25 = 3y - 18 + 96 - ²/₃(x + 27)
3x + ²/₃(x + 27) = 3y + 103
3x + ²/₃(x + 27) = 3(34) + 103
3x + ²/₃(x + 27) = 205
9x + 2x + 54 = 615
11x = 561
x = 561/11
x = 51
The value of angle 2 is calculated as;
∠2 = 180 - (3x - 25)
∠2 = 180 - 3x + 25
∠2 = 180 - 3(51) + 25
∠2 = 52⁰
The value of angle 1 is calculated as follows;
∠1 = 96 - ²/₃(x + 27)
∠1 = 96 - ²/₃(51 + 27)
∠1 = 44⁰
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(x + 1)(x-4) = 0
How would I solve this using the zero product properly
Answer:
The solutions to the equation (x + 1)(x - 4) = 0 are x = -1 and x = 4.
Step-by-step explanation:
To solve the equation (x + 1)(x - 4) = 0 using the zero product property, we need to set each factor equal to zero and solve for x:
x + 1 = 0 or x - 4 = 0
Solving for x in the first equation, we get:
x + 1 = 0
x = -1
Solving for x in the second equation, we get:
x - 4 = 0
x = 4
Therefore, the solutions to the equation (x + 1)(x - 4) = 0 are x = -1 and x = 4.
What’s the answer pls? I need help asap
The matrices which satisfy the condition AB ≠ BA are matrices A, B, C and F
What is a matrix?A matrix is an arrangement of objects in rows and columns.
Given the matrix A [tex]A = \left[\begin{array}{ccc}1&-2\\2&1\end{array}\right][/tex]we need to determine which of the following matrices could be used to determina if AB ≠ BA.
We proceed as follows
We know that matrix multiplication is not commutative. That is AB ≠ BA.
Now, the only condition for which it would be commutative is when AB = BA. This implies that B is either a zero matric or an identity matrix
So, the matrices which satisfy the condition AB ≠ BA are matrices A, B, C and F
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Convert the following function from vertex form to standard form. Show your work.
f(x) = 3(x — 8)^2 — 160
The required standard form of the function f(x) is 3x² - 48x + 32.
The vertex form of a quadratic function is:
f(x) = a(x - h)² + k
Where (h, k) is the vertex of the parabola and a is a coefficient that determines the shape of the parabola.
In the given function we can see that the vertex is at (8, -160) and a is 3. So we have:
f(x) = 3(x - 8)² - 160
Expanding the square:
f(x) = 3(x - 8)(x - 8) - 160
f(x) = 3(x² - 16x + 64) - 160
Distributing the 3,:
f(x) = 3x² - 48x + 192 - 160
Simplifying,
f(x) = 3x² - 48x + 32
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In a simple random sample, O the population is approximately normal with the mean, median, and mode being approximately equal O the mean is equal to the mode O the mean of N is always equal to the standard deviation O every member of the population has an equal chance of being selected
In a simple random sample, every member of the population has an equal chance of being selected. This sampling method helps ensure unbiased representation and allows for accurate statistical analysis.
In a simple random sample, every member of the population has an equal chance of being selected. This is one of the key characteristics of a simple random sample. Additionally, in a simple random sample, the population is assumed to be approximately normal with the mean, median, and mode being approximately equal. This assumption allows for easier analysis of the data collected from the sample. It is important to note that the mean of N (the sample size) is not always equal to the standard deviation. However, the mean of the sample can be used as an estimate of the population mean with a certain level of confidence.
In a simple random sample, every member of the population has an equal chance of being selected. This sampling method helps ensure unbiased representation and allows for accurate statistical analysis.
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please help with both! urgenttt
The value of sin Q is 3 / 5.
The length of AB in the right triangle is 10 units.
How to find the angles and side of a right triangle?A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.
Therefore, let's find the side and angles of the triangles.
sin Q = opposite / hypotenuse
sin Q = 9 / 15
sin Q = 3 / 5
Let's find the length of AB in the right triangle using Pythagoras's theorem,
8² + 6² = AB²
AB² = 64 + 36
AB = √100
AB = 10 units
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Hey! Can I get some help with this?
Answer: A 27√2
Step-by-step explanation:
You first need to simplify the roots
3√50+4√18
= [tex]3\sqrt{2*25} +4\sqrt{9*2}[/tex] break up what is under the root into factors of 50 and 18, make perfect squares is best.
For the first term, You can take the √25 =5 so 5 comes out and is multiplied by the 3 but the 2 stays unders the root
For the second term, you can take √9 =3 so 3 comes out and is multipled by the 4 but the 2 stays under the root.
=5*3√2 +4*3√2
=15√2 +12√2 now they are same terms so you can add
=27√2
To examine the effects of pets and friends in stressful situations, researchers recruited 45 people to participate in an experiment. Fifteen of the subjects were randomly assigned to each of three groups to perform a stressful task alone (control group), with a good friend present, or with their dog present. Each subject's mean heart rate during the task was recorded. Test the appropriate hypotheses at the α-0.05 level to decide if the mean heart rate differs between the groups
The p-value obtained from the ANOVA test is greater than 0.05, then there is not enough evidence to reject the null hypothesis, and it can be concluded that there is no significant difference in mean heart rate between the three groups.
The appropriate hypotheses to test are:
Null hypothesis (H0): The mean heart rate does not differ between the three groups performing the stressful task alone, with a friend present, or with their dog present.
Alternative hypothesis (Ha): The mean heart rate differs between at least two of the groups.
To test these hypotheses at the α-0.05 level, a one-way ANOVA (analysis of variance) test can be conducted. This test will help determine if there are significant differences between the means of the three groups.
If the p-value obtained from the ANOVA test is less than 0.05, then the null hypothesis can be rejected, indicating that there is a significant difference in mean heart rate between at least two of the groups. In this case, post-hoc tests such as Tukey's HSD (honest significant difference) test can be performed to determine which specific group means differ significantly.
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two common tolerance zones of a perpendicularity tolerance are two parallel planes or a cylinder. question 4 options: true false
The given statement "two common tolerance zones of a perpendicularity tolerance are two parallel planes or a cylinder" is true because Perpendicularity tolerance is a geometric tolerance that is used to control the orientation of a feature relative to a datum.
It ensures that a feature is perpendicular to a specified datum plane or axis within a certain tolerance zone. There are two types of tolerance zones for perpendicularity: a pair of parallel planes and a cylinder. The parallel plane tolerance zone consists of two parallel planes that are perpendicular to the datum plane or axis.
The feature must lie between these two planes within the specified tolerance limit. The cylinder tolerance zone, on the other hand, consists of a cylinder that is coaxial with the datum axis. The feature must lie within the cylinder and be perpendicular to the axis within the specified tolerance limit.
Perpendicularity tolerance is essential in ensuring the proper fit and function of parts in a machine or assembly. Without it, parts may not fit together properly, causing problems such as misalignment, binding, or excessive wear.
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Graph the solution to the following linear inequality in the coordinate plane. 2x+ygreater than 4
The graph of the inequality 2x + y > 4 in the coordinate plane is a shaded region above the line 2x + y = 4.
We have,
The given inequality is in the form of the equation of a straight line in two dimensions, but it also includes an inequality.
To graph this inequality on the coordinate plane, we first need to graph the equation of line 2x + y = 4.
We can do this by finding the x and y intercepts of the line.
To find the x-intercept, we set y = 0 and solve for x:
2x + 0 = 4
2x = 4
x = 2
So the x-intercept is (2, 0).
To find the y-intercept, we set x = 0 and solve for y:
2(0) + y = 4
y = 4
So the y-intercept is (0, 4).
Therefore,
The graph of the inequality 2x + y > 4 in the coordinate plane is a shaded region above the line 2x + y = 4.
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16. At Dunn's grocery store, small bas of goldfish crackers cost $3 and large bags cost $5. This month, if they sold a total of 280 bags of goldfish crackers and made $1,200; how many large bags did they sell?
Answer:
Let’s use algebra to solve this problem. Let x be the number of large bags sold, and y be the number of small bags sold. We know that:
x + y = 280 (total number of bags sold) 5x + 3y = 1200 (total revenue)
We can solve for x by multiplying the first equation by three and subtracting it from the second equation:
5x + 3y = 1200 -(3x + 3y = 840)
2x = 360 x = 180
Therefore, Dunn’s grocery store sold 180 large bags of goldfish crackers this month.
Step-by-step explanation:
Benjamin invests $400 in a savings account that earns 5% interest each
year.
Which expression gives the balance in the account after 3 years?
So first we need to find out what 5% of 400 is, we are gonna say we are on a calculator alright? So what we do is divide 0.05 by 400 and that equals 20. So after 3 years it would be 460$!
Please answer number 11.
Where the above parameters are given, Mikayla will need about 5,036.64 in² of paint to finish her project. Note that this is about surface area.
How to go about thisa) determine the surface area of each rectangular prism. The formula for this is:
SA = 2lw + 2lh + 2wh
Where I is the lenght, w is the width, and h is theheight.
For each of the prisms we have:
l = 3ft -= 36in
w = 2.5 ft = 30 inches
h = 3 inches
The surface area of the rectangular prism is:
= 2 (36 )(30) + 2 (36) ( 3) + 2 (30 ) (3)
= 2160 + 216 + 180
= 2556in²
B) Now it's time to compute the surface area of the culindrical hole using the formula:
SA = πdh
Recall that π = 3.14
Since d = 4 inches and
h = 3 inches
The surface area (SA) of the cylindrical hole is:
3.14 x 4 x 3
= 37.68in²
Thus, total surface area to be painted is:
2(2556 - 37.68)
= 2(2518.32)
= 5036.64in²
Thus, the assertion is correct that the total amount of paint required is approximately 5037 in² of paint.
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what is the ratio of the area of triangle bdc to the area of triangle adc? express your answer as a common fraction.
The ratio of the area of triangle BDC to the area of triangle ADC is
(√3 -1) : 2.
Angle ABC = 60°
Angle DBC = 45°
So, the measure of <ABD is
Angle ABD = ABC - DBC
= 60 - 45
= 15°
Now, the area of triangle CDB is
= (1/2) * S * BD * sin 45
and, the area of triangle ADB
= (1/2) * S * BD * sin 15
So, the area of triangle ADB to the area of triangle CDB =
= (1/2) * S * BD * sin 15 / (1/2 * S * BD * sin 45
= sin 15 / sin 45
Since, sin 15 = sin (45 -30)
= sin45 cos30 - sin 30 cos 45
= √2/2 √3/2 - (1/2)(√2/2)
= [√6 -√2] /4
= √2/2
So, sin 15 / sin 45 = [√6 -√2 ] / 4*2 /√2
= [ √6 - √2 ] / [ 2√2 ]
= √6 / [ 2√2 ] - √2 / [ 2√2]
= √3 / 2 - 1/ 2
= (√3 - 1 ) / 2
Thus, the ratio is (√3 -1) : 2.
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The question attached here seems to be incomplete, the complete question is:
Point D lies on side AC of equilateral triangle ABC such that the measure of angle DBC is 45 degrees. What is the ratio of the area of triangle ADB to the area of triangle CDB? Express your answer as a common fraction in simplest radical form.
Find the area•
Help?
Answer: 15
Step-by-step explanation:
Area of a parallelogram is base times height
A=bh b, base=5 h, height=3
A=(5)(3) =15 [tex]m^{2}[/tex]
10. Bonus The volume of each group of
solids is given below. Find the volume of
each of the 4 types of shapes.
na
V 8 π +8
V = 8 π + 28
V= 8 π +20
The volumes of the shapes are as follows: 33.14 cubic units , 55.14 cubic unit and 47.14 cubic units
How to determine the volumes of the shapes?
The given parameters are
a)
Volume (V) = 8π + 8
Where π = 22/7
Volume is = 8*22/7 +8
Volume = 176/7 + 8
V= 27.14 + 8
The volume is = 33.14 cubic units
b) Volume = 8π + 28
Where π = 22/7
By substitution we have
Volume is = 8*22/7 + 28
Volume = 176/7 + 28
V= 27.14 + 28
V= 55.14 cubic units
c) Volume = 8π + 20
Where π = 22/7
By substitution we have
Volume is = 8*22/7 + 20
Volume = 176/7 + 20
V= 27.14 + 20
V= 47.14 cubic units
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A 17 ft ladder leans against the side of a house. The bottom of the ladder is 7 ft away from the side of the house. Find x, the angle of elevation of the ladder.
Round your answer to the nearest tenth of a degree.
help please! :(
The angle of elevation of the ladder is 65.7 degrees.
How to find the side of a right triangle?A 17 ft ladder leans against the side of a house.
The bottom of the ladder is 7 ft away from the side of the house.
Let's find the angle of elevation of the ladder.
Therefore, let's use trigonometric ratios,
Hence,
cos x = adjacent / hypotenuse
adjacent = 7 units
hypotenuse = 17 units
cos x = 7 / 17
x = cos⁻¹ 0.41176470588
x = 65.6842611987
x = 65.7 degrees
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What is 1378129683.1232145345x1236451647.09872890001 divided by 12234345268.0910001928=______________________________________
The value of expression is,
⇒ 139,279,273.166959
We have to given that;
The expression is,
⇒ 1378129683.1232145345x1236451647.09872890001 divided by 12234345268.0910001928
Hence, We can simplify as;
⇒ ( 1378129683.1232145345 x 1236451647.09872890001 ) ÷ 12234345268.0910001928
⇒ 139,279,273.166959
Thus, The value of expression is,
⇒ 139,279,273.166959
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A florist is creating 10 centerpieces. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. A customer plans on spending $300 on 10 centerpieces with each centerpiece containing 12 flowers, with twice as many roses as the number of irises and lilies combined.
a) Write a system of linear equations that represents the situation.
b) Write a matrix equation that corresponds to your system.
c) Find the number of flowers of each type that the florist can use to create the 10 centerpieces using the inverse matrix (attached).
The florist may make 10 centerpieces with each centerpiece holding 12 flowers, costing the customer $300, using 1.5 roses, 3 lilies, and 1.5 irises.
a) Let's denote the number of roses, lilies, and irises in each centerpiece by r, l, and i, respectively. Then we have the following system of linear equations:
12 = r + l + i (since each centerpiece contains 12 flowers)
r = 2(i + l) (since there are twice as many roses as the number of irises and lilies combined)
2.5r + 4l + 2i = 30 (since the cost of each centerpiece is $30 and we are given the prices of each type of flower)
b) We may organize the variable coefficients into a matrix and the constants into a vector to get the matrix equation. The matrix equation will have the form Ax = b, where A is the coefficients matrix, x is the variable vector, and b is the constant vector. Here are the facts:
[tex]\begin{pmatrix}1 & 1 & 1 \\2 & -2 & -2 \\2.5 & 4 & 2 \\\end{pmatrix}\begin{pmatrix}r \\l \\i \\\end{pmatrix}$=\begin{pmatrix}12 \\0 \\30 \\\end{pmatrix}$[/tex]
c) We must solve the matrix equation using the inverse of the coefficient matrix in order to determine how many flowers of each kind the florist may utilize to build the 10 centerpieces. When we multiply both sides of the equation by the coefficient matrix's inverse, we obtain:
[tex]\begin{pmatrix}1 & 1 & 1 \\2 & -2 & -2 \\2.5 & 4 & 2 \\\end{pmatrix}^{-1} \begin{pmatrix}1 & 1 & 1 \\2 & -2 & -2 \\2.5 & 4 & 2 \\\end{pmatrix} \begin{pmatrix}r \\l \\i \\\end{pmatrix}=\begin{pmatrix}1 & 1 & 1 \\2 & -2 & -2 \\2.5 & 4 & 2 \\\end{pmatrix}^{-1}\begin{pmatrix}12 \\0 \\30 \\\end{pmatrix}[/tex]
Simplifying, we get:
[tex]\begin{pmatrix}r \\ l \\ i\\ \end{pmatrix}=\begin{pmatrix}1.5 \\3 \\1.5\\ \end{pmatrix}[/tex]
Therefore, the florist can use 1.5 roses, 3 lilies, and 1.5 irises in each centerpiece to create 10 centerpieces with each centerpiece containing 12 flowers, and the customer spending $300.
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