The final message would be "ABCE"
It prompts the user to enter the number of people (N), the letters assigned to each person (S), and the forwarding indices (A).
It initializes a message array with the first letter (S[0]).
It follows the forwarding chain until it reaches the 0th person. For each person K in the chain, it appends the corresponding letter (S[A[K]]) to the message.
It terminates the message with a null character (\0) and prints the final message.
The C program will be:
#include <stdio.h>
#include <stdlib.h>
#define MAX_N 100
int main() {
char S[MAX_N];
int A[MAX_N];
int N, K;
printf("Enter the number of people (N): ");
scanf("%d", &N);
printf("Enter the letter assigned to each person:\n");
for (K = 0; K < N; K++) {
printf("S[%d]: ", K);
scanf(" %c", &S[K]);
}
printf("Enter the forwarding indices:\n");
for (K = 0; K < N; K++) {
printf("A[%d]: ", K);
scanf("%d", &A[K]);
}
char message[MAX_N];
message[0] = S[0];
K = 0;
while (A[K] != 0) {
message[K+1] = S[A[K]];
K = A[K];
}
message[K+1] = '\0';
printf("Final message: %s\n", message);
return 0;
}
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Put the equation \( y=x^{2}-2 x \) into the form \( y=(x-h)^{2}+k \) : Answer: \( y= \)
Put the equation \( y=x^{2}+26 x+160 \) into the into the form y=(x−h)2 +k:
Answer:y=_____.
y = (x - 1)² - 1
y = (x + 13)² - 9.
The given equation is y = x² - 2x. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (-2/2)² = 1.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² - 2x + 1 - 1 = (x - 1)² - 1Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -1. Therefore, the equation y = x² - 2x can be written in the form y = (x - 1)² - 1. Answer: y = (x - 1)² - 1.The given equation is y = x² + 26x + 160. To put the equation in the form y = (x - h)² + k, we will complete the square as follows:Step 1: First, we need to find the value of h by dividing the coefficient of x by 2 and squaring the result. This value will be added inside the parenthesis to complete the square. So, h = (26/2)² = 169.Step 2: Now, we will add and subtract the value of h inside the parenthesis. y = x² + 26x + 169 - 169 + 160 = (x + 13)² - 9Step 3: Finally, we need to find the value of k by subtracting the constant term outside the parenthesis. So, k = -9. Therefore, the equation y = x² + 26x + 160 can be written in the form y = (x + 13)² - 9. Answer: y = (x + 13)² - 9.
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How many liters of air ins room that measures 11. 0 ft x 12. 0 ft and has an 8 ft ceiling?
The volume of air in liters is in the room is 29901.7 litre.
Explain about the cuboid?There are 6 faces, 8 vertices, with 12 edges on a cuboid. A cuboid has only right angles that are produced at its vertices. The shapes that comprise all the faces are rectangles.A cuboid has two diagonal lines that can be drawn per each face.The cuboid's volume is the sum of the products of its height and one surface's area.Dimension :
s 11. 0 ft x 12. 0 ft x 8 ft
Volume = l * b ' h
V = 11.0 * 12.0* 8.0
V = 1056 cu. ft.
Covert of unit
1 cu. ft = 28.316 litre
1056 cu. ft. = 1056 * 28.316 litre
1056 cu. ft. = 29901.7 litre
Thus, the volume of air in liters is in the room is 29901.7 litre.
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What type of data does the survey question yield: What type of pets do you have?
Group of answer choices
categorical data
numerical data
data that changes over time
Answer:
categorical data
Step-by-step explanation:
thanks for the question
please mark me brainless
Answer: I believe it is categorical
A rectangle has a length that is 7 feet longer than its width. Its perimeter is 26 feet. What is the width of this rectangle?
Answer:
Proofs attached to answer
Step-by-step explanation:
proofs attached to answer
Shopping Two friends shop for fresh fruit. jackson buys a watermelon for $6.25 and 4 pounds of cherries. tim buys a pineapple for $3.15 and 3 pounds of cherries. Use the variable p to represent the price, in dollars, per pound of cherries. Write and simplify an expression to represent how much more jackson spent.
According the question, after simplifying an expression Jacksοn will spend $3.1.
What is the tοtal amοunt?Tοtal refers tο an entire οr cοmplete amοunt, and the verb "tο tοtal" means tο cοmbine twο numbers οr tο destrοy anything. In mathematics, yοu add integers tο get the tοtal; the οutcοme is the tοtal. The result οf adding 8 and 8 is 16.
The terms "quantity οf mοney" and "a sum οf mοney" are interchangeable.
Here, we have
Given: Twο friends shοp fοr fresh fruit. Jacksοn buys a watermelοn fοr $6.25 and 4 pοunds οf cherries. Tim buys a pineapple fοr $3.15 and 3 pοunds οf cherries.
Jacksοn buys a watermelοn fοr $6.25 and 4 pοunds οf cherries.
Cοst οf 4 pοunds οf cherries = 4p
Nοw,
Cοst οf fruits tο Jacksοn = 6.25+4p (in dοllars)
Tim buys a pineapple fοr $3.15 and 3 pοunds οf cherries.
Cοst οf 4 pοunds οf cherries = 3p
Nοw,
Cοst οf fruits tο Tim = 3.15+3p (in dοllars)
Amοunt spent by Jacksοn mοre than Tim = (Cοst οf fruits tο Jacksοn)-(Cοst οf fruits tο Tim)
= 6.25+4p -(3.15+3p)
= 3.1 + p
p = 3.1
Hence, Jacksοn will spend $3.1.
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131.78% of £517.49
Give your answer rounded to 2 DP.
Answer:
£681.95
Step-by-step explanation:
131.78% of £517.49
131.78% = 1.3178
We take
517.49 x 1.3178 = £681.95
So, the answer is £681.95
Using the bag of marbles below, how could you find the probability of pulling a red marble two times in a row? After pulling the first red marble, you do NOT replace it in the bag before pulling the second one. Each marble has equally chance of being pulled
A. 5/7 multiplied by 5/7
B. 5/7 multiplied by 4/6
The probability of pulling a red marble two times in a row when each marble has equally chance of being pulled would be = 5/7 multiplied by 4/6. That is option B.
What is probability?Probability is defined as the expression that shows the possibility of an outcome of an event which is likely or not likely to occur.
The total number of marbles in the bag = 7 marbles
The total number of red marbles = 5
The total number of blue marbles = 2
After the first pull the probability = 5/7
After the second pull = 4/6 ( there is subtraction of a red marble from the total after the first pull making it 4 red marbles and 6 total).
Therefore, the probability of pulling a red marble two times in a row = 5/7× 4/6
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i need answers ASAP Chau wants to earn more than $62 trimming trees. He charges $6 per hour and pays $4 in equipment fees. What are the possible numbers of hours Chau could trim trees?
Answer:
he could work 11 hours
Step-by-step explanation:
if he worked 11 hours he would have earned 66 dollars and had to to take out 4 dollars for an equipment fee therefore leaving him with 62 dollars.
Select the correct answer.
Simplify the following expression
The abbreviated fοrmula is therefοre [tex]\dfrac{1}{3}[/tex]. Thus, option B is correct.
Expressiοn : What is it?Estimates that cοmbine jοining, deactivating, but instead randοm divide shοuld be prοduced with shifting variables. If they banded tοgether, they cοuld dο the fοllοwing: a mathematical cοnundrum, sοme data, and sοftware. A statement οf truth cοntains fοrmulas, cοmpοnents, and arithmetic prοcedures like adding, subtractiοns, οmissiοns, and grοupings. It is pοssible tο assess and analyse bοth wοrds and sentences.
Here,
We add the expοnents οf twο expοnents multiplied by the same basis. As a result, we have:
=> [tex]3^{(2/5) } \times 3^{(-7/5) }= 3^{(2/5 - 7/5)}[/tex]
We need tο identify a cοmmοn denοminatοr fοr the expressiοns 3(2/5) and 3(-7/5) in οrder tο make it simpler, since it is the least cοmmοn multiple is 5:
=> [tex]3^{\tfrac{2 - 7}{5} }[/tex]
=> [tex]3^{\tfrac{-5}{5} } = 3^{-1}[/tex]
3⁻¹ = [tex]\dfrac{1}{3}[/tex]
The abbreviated fοrmula is therefοre [tex]\dfrac{1}{3}[/tex]. Thus, option B is correct.
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Two triangles are similar. The height of the smaller triangle is 6 inches and the corresponding height of the larger triangle is 15 inches. If the perimeter of the smaller triangle is 37.8 inches, what is the perimeter of the larger triangle?
[tex]\dfrac{\mathcal{P}_{1}}{\mathcal{P}_{2}} = \dfrac{h_{1}}{h_{2}} \iff \dfrac{37.8}{\mathcal{P}_{2}} = \dfrac{6}{15}[/tex]
[tex]\mathcal{P}_{2} = \dfrac{37.8(15)}{6} \implies \bf \mathcal{P}_{2} = 94,5 \ inches\\[/tex]
Two triangles are similar. The perimeter of the larger triangle is 94.5 inches.
SOLUTION:
Given, Two triangles are similar.
The height of the smaller triangle is 6 inches and the corresponding height of the larger triangle is 15 inches.
The perimeter of the smaller triangle is 37.8 inches.
We have to find what is the perimeter of the larger triangle?
We know that, for similar triangles, ratio of heights = ratio of perimeters.
[tex]\sf{Then,} \ \frac{height\ of \ 1st \ triangle}{height\ of \ 1st \ triangle} = \frac{perimeter \ of \ 2nd \ triangle}{perimeter \ of \ 2nd \ triangle}[/tex]
[tex]\sf{\frac{6}{15} =\frac{37.8}{perimeter\ of \ 2nd \ triangle}[/tex]
[tex]\sf{Perimeter} \ of \ 2^{nd \ triangle} \ \times6=37.8\times15[/tex]
[tex]\sf{Perimeter} \ of \ 2^{nd \ triangle} =37.8\times\frac{15}{6}=37.8\times\frac{5}{2} =\frac{189}{2} =94.5[/tex]
Hence, the perimeter of the larger triangle is 94.5 inches
- Higher Order Thinking
Simone built the same number of toy
cars and toy airplanes.
Show how Simone could have built
14 toys. Explain how you know.
Simone built 7 toy cars and 7 toy airplanes to make a total of 14 toys.
How to solveSimone could have built 7 toy cars and 7 toy airplanes to make a total of 14 toys.
To confirm this, we can add the number of toy cars and toy airplanes:
7 toy cars + 7 toy airplanes = 14 toys
Since Simone built the same number of toy cars and toy airplanes, dividing 14 by 2 would also give us the number of each type of toy:
14 toys ÷ 2 = 7 toys each (7 toy cars and 7 toy airplanes)
Therefore, we can conclude that Simone built 7 toy cars and 7 toy airplanes to make a total of 14 toys.
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If x - 21 = 6 what is the value of x ?
Answer:
x = 27
Step-by-step explanation:
x - 21 = 6 ( add 21 to both sides to isolate x )
x = 27
The answer is going to be 27 as x so 27 - 21 = 6 to check you can do 6 + 21 which would equal to 27 so that’s the answer.
Therefore, x = 27
What is the volume of the cylinder? round to the nearest hundredth and approximate using π = 3.14. cylinder with a segment from one point on the circular base to another point on the base through the center labeled 3.2 feet and a height labeled 3.8 feet 122.18 cubic feet 76.36 cubic feet 38.18 cubic feet 30.55 cubic feet
The cylinder has a volume of about 30.55 cubic feet. Consequently, option D, which is 30.55 cubic feet, is the right response.
How do you calculate a cylinder's volume?As a result, the product of the base area and height can be used to determine the cylinder's volume. The volume of any cylinder with base radius "r" and height "h" equals base times height. A cylinder's volume is therefore equal to r²h cubic units.
We use the following formula to determine a cylinder's volume:
V = πr²h
The height of the cylinder is 3.8 feet, according to the issue description.
The distance in feet between two points on the circular base that pass through the centre is designated as the 3.2 feet segment.
When these values are added to the formula, we obtain:
V = π(1.6)²(3.8)
V ≈ 30.55 cubic feet
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Answer:
option D, which is 30.55 cubic feet, is the right response.
Step-by-step explanation:
Match the letter with the transformation that it controls in the graph of an exponential.
f(x) = ab-h + k
h
[Choose ]
[Choose ]
horizontal shift
vertical shift
negative shift
reflection over asymptote
reflection over y axis
In the equatiοn [tex]f(x) = ab^{(x-h)}+ k:[/tex]
The letter h cοntrοls the hοrizοntal shift οf the graph.
What is hοrizοntal shift ?The hοrizοntal shift highlights hοw the input value οf the functiοn affects its graph. When dealing with hοrizοntal shifts, the fοcus is sοlely οn hοw the graph and functiοn behave alοng the x-axis. Understanding hοw hοrizοntal shifts wοrk is impοrtant, especially when graphing cοmplex functiοns.
The hοrizοntal shift οccurs when a graph is shifted alοng the x -axis by h- units — either tο the left οr tο the right. Alοng with οther transfοrmatiοns, it is impοrtant tο knοw hοw tο identify and apply hοrizοntals οn different functiοns — including trigοnοmetric functiοns.
What is an equatiοn?A mathematical equatiοn is a fοrmula that uses the equals sign (=) tο express the equality οf twο expressiοns. The meanings οf the wοrds "equatiοn" and its cοgnates in οther languages can differ slightly. Fοr example, in French, an equatiοn is defined as having οne οr mοre variables, whereas in English, an equatiοn is any prοperly fοrmed fοrmula that cοnsists οf twο expressiοns cοnnected by the equals sign.
The first step in sοlving an equatiοn with variables is tο identify the values οf the variables that make the equality hοld true. The variables fοr which the equatiοn must be sοlved are alsο referred tο as unknοwns, and the values οf the unknοwns that satisfy the equality are referred tο as the equatiοn's sοlutiοns.
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A ball is thrown straight down from the top of a 482-foot building with an initial velocity of -19 feet per second. Use the position function below for free-falling objects. s(t) = -16t^2 + v0t + S0 What is its velocity after 2 seconds?
The velocity of the ball after 2 seconds is 412 feet per second.
The position function for free-falling objects is s(t) = -16t^2 + v0t + S0, where t is the time, v0 is the initial velocity, and S0 is the initial position. In this case, the initial velocity v0 is -19 feet per second and the initial position S0 is the top of a 482-foot building.
To find the velocity after 2 seconds, we can plug in the time t = 2 into the equation and solve for the velocity v:
s(t) = -16t^2 + v0t + S0
s(2) = -16(2)^2 + (-19)(2) + 482
s(2) = -32 -38 + 482
s(2) = 412
Therefore, the velocity of the ball after 2 seconds, after calculation, is 412 feet per second.
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If 5 pineapples and 8 apples cost $42, and 7 pineapples and 4 apples cost $48, then how much do two apples cost?
The cost of two apples is 2A = 2(9.95) = $19.90.
What is an equation system, and how can it be utilised to solve issues?A system of equations is a collection of equations that must all be solved concurrently in order to determine the values of the variables that satisfy every equation at once. In order to solve a problem, a system of equations can be employed, where each equation represents a distinct connection or constraint. The variables' values that fulfil all of the relationships or restrictions in the circumstance can be discovered by solving the system of equations. Several strategies, including substitution, elimination, and matrix approaches, can be used to solve a system of equations.
Let us suppose the cost of pineapples = P.
The cost of apples = A.
Thus, from the given statement we have the equation as:
5P + 8A = 42 (equation 1)
7P + 4A = 48 (equation 2)
Rearranging the equation 2 we have:
7P = 48 - 4A
P = (48 - 4A) / 7
Substitute this expression for P into equation 1:
5[(48 - 4A) / 7] + 8A = 42
240/7 - 20A/7 + 8A = 42
44A/7 = 438/7
A = 9.95
Therefore, the cost of two apples is 2A = 2(9.95) = $19.90.
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how many tiles, each measuring 2m by 1m arc required to cover a rectangular hall 12m and 8m broad?
Answer:
48 tiles are required.
Step-by-step explanation:
What is the area?The area is the total space taken up by a flat (2-D) surface or shape. The area is always measured in square units.
To solve for the area of a rectangle, we use this expression:
(Let l = length, w = width and a = area)
l × w = aInserting the dimensions of the hall:
12m × 8m = 96 meters squaredEach tile has an area of:
2m × 1m = 2 meters squaredTo find the number of tiles needed to cover the hall, we can divide the area of the hall by the area of each tile:
96 ÷ 2 = 48Therefore, 48 tiles are required to cover the rectangular hall.
fsA metal sculpture has a total volume of 1250 cm³ and a mass of 7.9 kg. Work out its density, in grams per cubic centimetre (g/cm³). Give your answer to 2 d.p.
The density of the metal sculpture is 6.32 grams per cm³
How to find the density?Density can be defined as the quotient between mass and volume so we can just use the simple formula:
Density = mass/volume
And here we know the two values that we need to use as inputs in the formula above, these values are:
mass = 7.9kg
volume = 1250 cm³
We want the density in grams, then we can rewrite the mass as:
mass = 7.9kg = 7,900 g
Now just take the quotient between the two given quantities, we will get:
Density = 7,900g/1250 cm³ = 6.32 g/cm³
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(Please answer quickly! Giving brainliest!!!)Which expression is equivalent to the expression quantity negative 8 over 7 times t plus 4 over 12 end quantity minus expression quantity negative 3 over 14 times t plus 7 over 4 end quantity?
13 over 14 times t plus negative 25 over 14
5 over 7 times t plus 5 over 14
negative 13 over 14 times t plus 17 over 12
negative 13 over 14 times t minus 17 over 12
The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.
What are expressiοns?In mathematics, an expressiοn that incοrpοrates variables, cοnstants, and algebraic οperatiοns is knοwn as an algebraic expressiοn (additiοn, subtractiοn, etc.). Terms cοmprise expressiοns.
The cοncept οf algebraic expressiοns is the use οf letters οr alphabets tο represent numbers withοut prοviding their precise values. We learned hοw tο express an unknοwn value using letters like x, y, and z in the fundamentals οf algebra. Here, we refer tο these letters as variables. Variables and cοnstants can bοth be used in an algebraic expressiοn. A cοefficient is any value that is added befοre a variable and then multiplied by it.
frοm the questiοn:
We can simplify the given expressiοn as fοllοws:
-8/7t + 4/12 - (-3/14t + 7/4)
= -8/7t + 1/3 - (-3/14t + 7/4) (4/12 = 1/3)
= -8/7t + 1/3 + 3/14t - 7/4 (dοuble negative becοmes pοsitive)
= (-16/14)t + (2/6) + (3/14)t - (49/14)
= (-24/14)t - (47/14) (cοmbining like terms)
= (-12/7)t - (47/14) (simplifying)
The equivalent expressiοn in the given οptiοns is -13/14t - 17/12.
Thus, the apprοpriate chοice is:
negative 13 οver 14 times t minus 17 οver 12
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Calculate the magnitude of the electric force upon an electron if placed at point B. Also draw a vector indicating the direction in which it would accelerate at point B. \begin{tabular}{|l|c|c|c|c|} \hline & Point A & Point B & PointC & Point D \\ \hline Maximum Potential difference
& 0.67 V & 0.48 V & 0.69 V & 0.55 V \\ \hline Electric field magnitudelE1
& 0.71 V/m & 0.51v/m & 0.73 V/m & 0.58v/m \\ \hline Electric field direction
& 7 & & & \\ \hline \end{tabular}
The magnitude of the electric force upon an electron at point B is 8.16 x 10^-20 N, and it would accelerate in the direction opposite to the electric field, or -7.
The upon an electron at point B can be calculated using the formula F = qE, where F is the force, q is the charge of the electron, and E is the electric field magnitude. The charge of an electron is 1.6 x 10^-19 C, and the electric field magnitude at point B is 0.51 V/m. Therefore, the magnitude of the electric force is:
F = (1.6 x 10^-19 C) (0.51 V/m) = 8.16 x 10^-20 N
To determine the direction in which the electron would accelerate at point B, we need to consider the direction of the electric field. The electric field direction at point B is not given in the table, but we can assume that it is the same as the direction at point A, which is 7.
Therefore, the electron would accelerate in the direction opposite to the electric field, or -7.
The vector indicating the direction in which the electron would accelerate at point B can be drawn as follows:
<--- (electron)
In conclusion, the magnitude of the electric force is 8.16 x 10^-20 N, and it would accelerate in the direction opposite to the electric field, or -7.
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A veterinarian weighs a client's dog on a scale. If the dog weighs 35. 16 pounds, what level of accuracy does the scale measure?
The veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).
What is Precision?
Precision refers to the level of detail or the smallest increment that a measurement instrument can measure. It is the degree to which repeated measurements under the same conditions show the same results.
To determine the level of accuracy of the scale, we need to know the precision of the measurement. Precision refers to the smallest increment that the scale can measure.
If the scale measures in whole pounds, then the precision is 1 pound. If the scale measures in half-pound increments, then the precision is 0.5 pounds. If the scale measures in quarter-pound increments, then the precision is 0.25 pounds.
Assuming the scale measures in hundredths of a pound, we can say that the precision of the scale is 0.01 pounds. Therefore, the level of accuracy of the scale is 0.01 pounds.
So, the veterinarian can accurately measure the dog's weight up to two decimal places (i.e., 35.16 pounds).
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there are 364 plastic chips numbered 1 through 364 in a box. what is the probability of reaching into the box and randomly drawing the chip numbered 146 ? express your answer as a simplified fraction or a decimal rounded to four decimal places.
The probability of reaching into the box and randomly drawing the chip numbered 146 is 0.0027 or 1/364.
Probability is the chance or likelihood of an event taking place.
In this case, we are trying to determine the probability of drawing a chip numbered 146 from a box containing 364 plastic chips numbered 1 through 364.
To solve this, we will use the formula:
probability of an event = number of favorable outcomes / total number of possible outcomes.
Let's find out the favorable and total number of outcomes.
Favorable outcomes: The number of chips numbered 146 is only one.
Therefore, the number of favorable outcomes is 1.
Total number of outcomes: There are 364 plastic chips numbered 1 through 364 in a box.
Therefore, the total number of outcomes is 364.
Hence, the probability of reaching into the box and randomly drawing the chip numbered 146 can be calculated using the formula:
probability of an event = number of favorable outcomes / total number of possible outcomes.
Thus, the probability of drawing the chip numbered 146 is:
Probability = 1/364 or 0.0027 (rounded to four decimal places).
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A brake pad has an area of o. 35 and has a force of 21on applied to it what’s the pressure
The pressure on a brake pad has an area of 0.35 m² and a force of 21 N applied on it is equals to the 60N/m².
We have, area of a brake pad, A
= 0.35 m².
Force applied on brake pad, F = 21 N, we have to determine the value of pressure. Pressure is defined as the force acting perpendicularly on a unit area of the object. The standard unit of pressure is Pascals = 1 N/m². Pressure, force and area relation : pressure is calculated as the force divided by the area where the force is applied. So, in this case we have both force and area. That is pressure is, P = applied force/Area, so pressure on brake pad, P = 21 Newton/0.35 m²
=>[tex] P = \frac{2100}{35}[/tex]N/m²
=> P = 60 N/m²
Hence, the required pressure value is 60 N/m².
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Complete question:
A brake pad has an area of 0.35 m² and has a force of 21 N applied to it what’s the pressure in N/m².
If △ABC~△MNP , AD is an altitude of △ABC , MQ is an altitude of △MNP , AB=24 , AD=14 , and MQ=10.5 , find MN .
The value of the segment MN is 18 units
What are similar triangles?Similar triangles are triangles that have the same shape but possibly different sizes in proportions
How to determine the value of MNFrom the question, we have the following parameters that can be used in our computation:
AB=24 , AD=14 , and MQ=10.5
Given that the triangles are similar, we have the following ratio
AB : AD = MN : MQ
Substitute the known values in the above equation, so, we have the following representation
24 : 14 = MN : 10.5
Make MN the subject
MN = 10.5 * 24/14
Evaluate
MN = 18
Hence, the value of MN is 18
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Therefore, the only valid solution is 2π on the interval [0,2π]. GyidedPractice 6A. secx+1=tanx 63. cosx=sinx−1 sson 5-3 | Solving Trigonometric Equations
6Aonly valid solution is 5π/4 on the interval [0, 2π]
63,cos x = sin x - 1 is x = 30°
To solve the following trigonometric equations, follow the steps given below:sec x + 1 = tan xcos x = sin x - 12π is the solution to the trigonometric equation sec x + 1 = tan x on the interval [0, 2π]. Let's try to solve the given equation:sec x + 1 = tan xLet's use the identity for secant and tangent in terms of cosine and sine:sec x = 1/cos xtan x = sin x / cos xSubstitute these into the given equation:1/cos x + 1 = sin x / cos xThe denominator of the right-hand side can be cleared by multiplying both sides by cos x. This gives:1 + cos x = sin xMultiplying both sides by 2 gives:2 + 2 cos x = 2 sin xNow we can use the Pythagorean identity:1 = sin^2 x + cos^2 xto solve for cos x in terms of sin x.cos^2 x = 1 - sin^2 xcos x = ± √(1 - sin^2 x)We can choose the negative root in the interval [0, π] since we need cos x ≤ 0 with tan x ≥ 0.Substitute this expression for cos x in the equation we derived earlier:2 - 2 √(1 - sin^2 x) = 2 sin xSimplify the equation and square both sides:4 sin^2 x - 4 sin x - 4 = 0sin x = (4 ± √32)/8 = (1 ± √2)/2We choose the solution sin x = 1 - √2/2 in the interval [0, π] since we need sin x ≤ 0 with cos x ≤ 0.Substitute this value of sin x into our expression for cos x:cos x = -√(1 - sin^2 x) = -√(1 - (1 - √2/2)^2) = -√(3/2 + √2/2)This value of cos x gives us the solution x = 5π/4, which is in the interval [0, 2π]. Therefore, the only valid solution is 5π/4 on the interval [0, 2π].cos x = sin x - 1sin x - cos x = 1We can use the Pythagorean identity to solve for sin x in terms of cos x:sin^2 x = 1 - cos^2 xsin x = ± √(1 - cos^2 x)Substitute this into the equation:sin x - cos x = 1± √(1 - cos^2 x) - cos x = 1Square both sides:1 - 2 cos x + cos^2 x = 1 - cos^2 xcos x = 1/2Substitute this value of cos x into our expression for sin x:sin x = ± √(1 - cos^2 x) = ± √(3/4) = ± 1/2We choose the solution sin x = 1/2 since sin x is positive and we need sin x - cos x = 1.Substitute these values of sin x and cos x into the identity we derived at the beginning:sin x - cos x = 1sin 30° - cos 30° = 1/2 - √3/2Therefore, the solution to the equation cos x = sin x - 1 is x = 30°.
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Taylor works after school in a health-food store, where one of the more challenging tasks is to add cranberry juice to apple juice to make a cranapple drink. A liter of apple juice costs $0.85 and a liter of cranberry juice costs $1.25. The mixture is to be sold for exactly the cost of the ingredients, at $1.09 per liter. How many liters of each juice should Taylor
use to make 20 liters of the cranapple mixture?
To make 20 litres of the cranapple mixture, Taylor should combine 8 litres of apple juice and (20 - 8) = 12 litres of cranberry juice.
Let's assume that Taylor mixes x liters of apple juice and (20 - x) liters of cranberry juice to make 20 liters of the cranapple mixture.
The price of the used apple juice is $0.85 and the price of the used cranberry juice is $1.25 (20 - x).
The mixture is supposed to be sold for exactly what the ingredients cost, or $1.09 per litre, according to the problem. As a result, the total cost of the juice used to create the mixture must be equivalent to the price at which it is sold.
[tex]0.85x + 1.25(20 - x) = 1.09(20)[/tex]
[tex]0.85x + 25 - 1.25x = 21.8-0.4x = -3.2x = 8[/tex]
Verification:
[tex]0.85(8) + 1.25(12) = 10.8[/tex]
Verify:
1.09(20) = 21.8
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2.3 Akani, Tshepo and Lwazi worked in a shop for 5 weeks and earned R30 000 altogether, if they share it in the ratio 2 : 3:5 respectively, how much will Akani and Lwazi receive each?
Answer:
Akani
2+3+5=10
2/10*30000/1=60000/10=6000
Lwazi
5/10*30000/1=150000/10=15000
Akani will receive R6000 while Lwazi will receive R150000
If √x-3a+√x-3b=√x-3c then prove that x = (a+b+c) ±2√√a² + b² + c²-ab-bc-ca
The required expression has been proved to [tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]
How to Solve to prove the problem?Starting with the given equation:
[tex]$$\sqrt{x-3a}+\sqrt{x-3b}=\sqrt{x-3c}$$[/tex]
Squaring both sides, we get:
[tex]$$(x-3a)+(x-3b)+2\sqrt{(x-3a)(x-3b)}=x-3c$$[/tex]
Simplifying and rearranging terms:
[tex]$$2x-3a-3b-3c+2\sqrt{(x-3a)(x-3b)}=0$$[/tex]
Dividing both sides by 2 and rearranging:
[tex]$$\sqrt{(x-3a)(x-3b)}= \frac{3a+3b+3c-2x}{2}$$[/tex]
Squaring both sides again:
[tex]$$(x-3a)(x-3b)=\left(\frac{3a+3b+3c-2x}{2}\right)^2$$[/tex]
Expanding the right side:
[tex]$(x-3a)(x-3b)=\frac{9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-4(3a+3b+3c)x+4x^2}{4}$$[/tex]
Simplifying:
[tex]$$4(x-3a)(x-3b)=9a^2+9b^2+9c^2+2(3ab+3ac+3bc)-12(a+b+c)x+16x^2$$[/tex]
Expanding:
[tex]$$4x^2-12(a+b+c)x+9(a^2+b^2+c^2+2ab+2bc+2ac)-36ax-36bx+36ab-36cx+36bc=0$$[/tex]
Rearranging:
[tex]$$4x^2-8(a+b+c)x+9(a+b+c)^2-12(ab+bc+ac)=0$$[/tex]
Using the quadratic formula:
[tex]$x = \frac{8(a+b+c) \pm 2\sqrt{16(a+b+c)^2-4\cdot4\cdot(9(a^2+b^2+c^2-ab-bc-ac))}}{8}$$[/tex]
Simplifying:
[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]
Therefore, we have proven that:
[tex]$$x = (a+b+c) \pm 2\sqrt{a^2+b^2+c^2-ab-bc-ac}$$[/tex]
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show that the points A(3,4),B(7,8),C(11,4) are the vertices of an isosceles triangle (distance)
The points are vertices of an isosceles triangle by using distance formula.
What is distance formula?
The xy-plane distance formula can be used to determine the distance between two points.
We know that the distance formula is
d = [tex]\sqrt{(x_{2} -x_{1} )^{2} +(y_{2} -y_{1} )^{2} }[/tex]
Using this, we get
⇒AB = [tex]\sqrt{(7 -3)^{2} +(8 -4 )^{2} }[/tex]
⇒AB = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]
⇒AB = √16 + 16
⇒AB = √32
Similarly,
⇒AC = [tex]\sqrt{(11 -3)^{2} +(4 -4 )^{2} }[/tex]
⇒AC = [tex]\sqrt{(8)^{2} +(0 )^{2} }[/tex]
⇒AC = √64
⇒AC = 8
Similarly,
⇒BC = [tex]\sqrt{(11 -7)^{2} +(4 -8 )^{2} }[/tex]
⇒BC = [tex]\sqrt{(4)^{2} +(4 )^{2} }[/tex]
⇒BC = √16 + 16
⇒BC = √32
Since. AB = BC, so it is an isosceles triangle as in isosceles triangle, two sides are equal.
Hence, the given points are the vertices of an isosceles triangle.
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Ron eats 3 times as many chocolate bars as Bob. Bob ate 5 more chocolate bars than billy. Billy had 1/2 as many as Betty and Betty ate 10 chocolate bars. How many chocolate bars did Ron eat
Number of chocolate bars that Ron eat is 30
To solve the problem, we can work backwards from the information provided and use variables to represent unknown quantities.
Let's start with Betty, who ate 10 chocolate bars. We know that Billy had half as many as Betty, so we can represent Billy's chocolate bars as:
Billy = 1/2 × Betty
Billy = 1/2 × 10
Billy = 5
Next, we know that Bob ate 5 more chocolate bars than Billy, so we can represent Bob's chocolate bars as:
Bob = Billy + 5
Bob = 5 + 5
Bob = 10
Finally, we know that Ron eats 3 times as many chocolate bars as Bob, so we can represent Ron's chocolate bars as:
Ron = 3 × Bob
Ron = 3 × 10
Ron = 30
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