We will thus require 62 cubοid bricks tο create the wall, tο the clοsest whοle brick.
What shape dοes a brick have?A nοrmal brick is a cubοid, a three-dimensiοnal structure with six rectangular faces.
We need tο have at minimum 42/18 = 2.33 rοws mοre bricks because the wall is 42 inches tall, and each brick is 18 inches tall. We require three rοws οf bricks since we can't have a half rοw.
We require 144/7 = 20.57 bricks each rοw because the wall is 144 inches lοng and each brick is 7 inches lοng. There must be 21 bricks each rοw because we cannοt have half bricks.
The first rοw requires 21 whοle bricks. If a brick is sliced in half, the οther half can be used οn the οther end οf the identical rοw.
21 whοle bricks are alsο required fοr the secοnd rοw. Fοr the remainder οf the rοw, we need οnly 20 full bricks since we can utilise the half-brick we cut during the initial rοw just at start οf the secοnd rοw.
The third rοw requires 21 whοle bricks. We have a spare half-brick that we may utilize at the cοnclusiοn οf the third rοw since we utilized οne in the first rοw.
Hence, 62 cοmplete bricks are needed, which equals 21 + 20 + 21.
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On Saturday mornings, Ronald volunteers at the hospital where his mother works. One Saturday, he answers phone calls at the information desk while the receptionist is away. Then he spends 40 minutes delivering flowers to patients' rooms. In all, Ronald volunteers at the hospital for 90 minutes that day. Which equation can you use to find the amount of time , that Ronald answers phone calls?
Ronald spent 50 minutes answering phone calls on Saturday morning. The equation used to find this value is x + 40 = 90.
Let's assume that Ronald spent "x" minutes answering phone calls. We know that he spent a total of 90 minutes volunteering, and 40 minutes delivering flowers. So the time he spent answering phone calls and delivering flowers can be expressed as:
x + 40
We also know that the total time he spent volunteering was 90 minutes. So we can write:
x + 40 = 90
To solve for "x", we can subtract 40 from both sides of the equation:
x + 40 - 40 = 90 - 40
Simplifying:
x = 50
where x represents the amount of time Ronald spent answering phone calls.
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Lily is practicing multiplying complex numbers using the complex number (2+i).
To determine the value of (2+i)²,Lily performs the following operations:
Step 1: (2+i)²= 4+i²
Step 2: 4+i² = 4+ (−1)
Step 3: 4+(-1)=3
Lily made an error.
Explain Lily's error and correct the step which contains the error.
Bonus:
Lily is continuing to explore different ways in which complex numbers can be multiplied so the
answer is not a complex number. Lily multiplies (2+i) and (a+bi), where a and b are real
numbers, and finds that her answer is not a complex number.
A. Write an equation that expresses the relationship between a and b.
Lily's error is in the first step, (2+i)^2 ≠ 4 + i^2.
(2+i)^2 = (2+i)(2+i) and you need to FOIL.
(2+i)^2 = (2+i)(2+i)
= 4 + 2i + 2i + i^2
= 4 + 4i + (-1)
= 3 + 4i
Bonus:
If you want to multipy (2+i) by (a+bi) and not end up with a complex number, you'd first FOIL
(2+i)(a+bi) = 2a + 2bi + ai + bi^2
We know i^2 = -1, so this becomes
= 2a + 2bi + ai + b(-1)
= 2a + 2bi + ai - b
= 2a - b + 2bi + ai
Now, for this not to be complex, we need the imaginary pieces to cancel each other out. In other words 2bi+ai=0. For that to happen, 2b + a = 0, or
2b = -a or a = -2b
So it would seem that if we pick any b-value and make a = -2b, then we'll end up with a non-complex number.
Let's try b=5, making a = -10
(2+i)(-10+5i) = -20 + 10i - 10i + 5i^2
The 10i's cancel and 5i^2 = -5, so we're left just with -25.
Find the indicated real nth roots of a n=3, a=-125
The cube root of -1 is -1, and the cube root ∛(-125) is -5.
To find the real nth roots of a number a, we can use the formula:
[tex]√(a) = a^(1/n)[/tex]
For the case where n=3 and a=-125, we have:
[tex]√(-125) = (-125)^(1/3)[/tex]
We can simplify this using the fact that
[tex](-a)^(1/n) = -(a^(1/n)):[/tex]
√(-125) = - (√125)
Now we need to find the cube root of 125. We can factor 125 as 555, so:
√(-125) = - (√125) = - (√(555)) = - (5√5)
Therefore, the real cube root of -125 is -5√5.
The cube root of -125 is the real number x that satisfies the equation
[tex] {x}^{3} [/tex]
= -125.
We can rewrite -125 as -1*5^3, which means we can write the cube root of -125 as ∛(-1)*∛(5^3).
The cube root of -1 is -1, and the cube root of 5^3 is 5, so ∛(-125) = -1*5 = -5.
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A farmer has built circular pens for her animals behind the barn.
She needs to determine how far the center of each pen is from the barn in order to install
underground wiring for lights.
Here's what she knows:
The distance between the center of the cow pen and the center of the horse pen is 15 yards.
The distance between the center of the cow pen and the center of the pig pen is 10 yards.
All radii are whole numbers, and all are greater than 3 yards
No two pens are the same size.
She has labeled distances between the barn & points of tangency to the pens in her diagram.
You can help her decide which animal goes in which pen.
1. Now help her find the distance from the barn to the center of each pen! Show all of your work.
Use the page below for workspace as needed.
Distance from barn to center of...
Horse pen:
Cow pen:
Pig pen:
18.25
9.5 yds
+
75 yds.
Pig
COW
112248
8.75 yds.
24.75
horse
16 yds.
The distances from the barn to the center of each pen are: Horse pen: 47 yards , Cow pen: 50.25 yards , Pig pen: 64.75 yards.
Define Pythagorean theorem?The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
To solve for the distances from the barn to the center of each pen, we can use the following steps:
Draw a diagram of the situation, labeling the radii of each pen and the distances between the pens.
Use the Pythagorean theorem to solve for the radius of each pen. Let r1 be the radius of the horse pen, r2 be the radius of the cow pen, and r3 be the radius of the pig pen.
For the horse pen:
r1² = (15 + 16)² + (18)²
r1²= 961
r1 = 31 (rounded to the nearest whole number)
For the pig pen:
r3² = (10 + 24.75)² + (18)²
r3²= 1640.06
r3 = 40 (rounded to the nearest whole number)
Use the distance from the barn to the point of tangency of each pen to solve for the distance from the barn to the center of each pen. Let d1 be the distance from the barn to the center of the horse pen, d2 be the distance from the barn to the center of the cow pen, and d3 be the distance from the barn to the center of the pig pen.
For the horse pen:
d1 = 16 + r1
d1 = 16 + 31
d1 = 47 yards
For the cow pen:
d2 = 75 - r2
d2 = 75 - 24.75
d2 = 50.25 yards
For the pig pen:
d3 = 24.75 + r3
d3 = 24.75 + 40
d3 = 64.75 yards
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The pictures show four wavelengths of sound.
on
wo
W
X
LO
commo
то
wwwwwww.
Y
O
Z
Which line shows the lowest frequency?
The line labelled “W” shows the lowest frequency of sound.
What is sound?Sound is a type of energy that is created by vibrations and travels through the air in the form of waves. Sound can only be heard when these waves enter the ear and vibrate the eardrum. The frequency of the sound wave determines the pitch of the sound. Sound is also used to communicate and can be manipulated to create music. It can be used to detect events and can travel through water and solid objects.
Frequency is a measure of how often a sound wave repeats itself over time. It is measured in hertz (Hz), which is the number of sound waves per second. The lower the frequency of a sound wave, the lower the pitch of the sound. The line labelled “W” shows the lowest frequency of sound, meaning it has the lowest pitch. The other lines show higher frequencies, meaning they have higher pitches.
The other lines show different frequencies of sound, each with its own pitch. Line “X” has a higher frequency than line “W”, but still lower than the other lines. Line “LO” has a higher frequency than “X”, but lower than the other lines. Line “Y” has a higher frequency than “LO”, but still lower than the other lines. Line “O” has the highest frequency and therefore the highest pitch among all of the lines.
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Finding values of an Inverse Using a Table values. =.31.f−1(5) 34. f(f−1(6)) ax. f−1(f(1)) 36. f−1(f−1(0)) 37−48= Inverse Function Property we une inverse Funge . Property to show that f and g are inverses of each other. 37. f(x)=x−6:g(x)=x+6 38. f(x)=3x,g(x)=3x e.39. f(x)=3x+4;g(x)=3x−4 40. f(x)=2−5x;g(x)=52−x 41. f(x)=x1;g(x)=x1 42. f(x)=x5;g(x)=5x
43. f(x)=x2−9,x≥0;g(x)=x+9
,x≥−9 44. f(x)=x3+1;g(x)=(x−1)1/3 45. f(x)=x−11;g(x)=x1+1
31.f(f-1(6)) = 6
34.f-1(f(1)) = 1
36.f-1(f-1(0)) = 0
37.f(g(x)) = f(x + 6) = x + 6 - 6 = x g(f(x)) = g(x - 6) = x - 6 + 6 = x
39.f(g(x)) = f(3x) = 3(3x) = x g(f(x)) = g(3x) = 3(3x) = x
40.f(g(x)) = f(3x - 4) = 3(3x - 4) + 4 = x g(f(x)) = g(3x + 4) = 3(3x + 4) - 4 = x
41. f(g(x)) = f(5/2 - x) = 2 - 5(5/2 - x) = x g(f(x)) = g(2 - 5x) = 5/2 - (2 - 5x) = x
42. f(g(x)) = f(x2) = (x2)1/2 = x g(f(x)) = g(x1/2) = (x1/2)2 = x
43.f(g(x)) = f(x + 9) = (x + 9)2 - 9 = x g(f(x)) = g(x2 - 9) = x2 - 9 + 9 = x
44.f(g(x)) = f((x - 1)1/3) = (x - 1)1/33 + 1 = x g(f(x)) = g(x3 + 1) = (x3 + 1 - 1)1/3 = x
45.f(g(x)) = f(x1/2 + 1) = x1/2 + 1 - 11 = x g(f(x)) = g(x - 11) = (x - 11)1/2 + 1 = x
To answer your questions, here is the following information:
1. Finding values of an Inverse Using a Table values: f-1(5) = 0.31
f(f-1(6)) = 6
f-1(f(1)) = 1
f-1(f-1(0)) = 0
2. Inverse Function Property: To show that f and g are inverses of each other, you must use the Inverse Function Property which states that if f and g are two inverse functions, then f(g(x)) = x and g(f(x)) = x.
3. Examples:
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Robyn tossed a two-color chip counter 3 times. One side of the chip counter is red, the other side is yellow. Which tree diagram shows all the possible outcomes, where R represents red and Y represents yellow?
In this case, there are 2 outcomes for each toss, so there are 2³ = 8 possible outcomes in total.
What is tree diagram?A tree diagram is a visual tool used to represent a set of possible outcomes or decisions in a systematic way, branching out from a central point.
Here is the tree diagram showing all the possible outcomes for tossing a two-color chip counter three times:
R Y
/ | \
R R Y
/ \ / \ /
R Y R Y R Y
Each branch represents a single toss of the chip counter, with the two possible outcomes represented by the two branches extending from each node.
The outcome of each toss is independent of the outcome of the other tosses, so the total number of possible outcomes is the product of the number of outcomes for each toss.
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Alisa is choosing new tile for the floor in her dining room which is in the shape of a square with side lengths x feet. The tile costs $3. 39 per square foot
Th function is f(x) = 3.39x²
The cost of the flooring for a dining room is $410.19.
The cost of the flooring for a dining room is $6,072.31.
What is a function?
A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
Area of a square = side²
Square with side lengths = x feet.
The tile costs = $3.39 per square foot.
a.
The function f for the cost of the flooring.
f(x) = 3.39x²
b.
x = 11 feet
f(11) = 3.39 x 11²
f(11) = 3.39 x 121
f(11) = $410.19
c.
x = 23 feet
f(23) = 3.39 x 23²
f(23) = 3.39 x 529
f(23) = $6,072.31
Therefore,
f(x) = 3.39x²
The cost of the flooring for a dining room is $410.19.
The cost of the flooring for a dining room is $6,072.31.
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Complete Question
Alisa is choosing new tile for the floor in her dining room which is in the shape of a square with side lengths x feet. The tile costs $3.39 per square foot.
a. Write the function f for the cost of the flooring.
b. Determine the cost of the flooring if she decides on a dining room with side lengths of 11 ft.
c. Determine the cost of the flooring if she decides on a dining room with side lengths of 23 ft
Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. About ______% of the area is between z=−3.5 and z=3.5 (or within 3.5 standard deviations of the mean).
99.96% of area lies between given value range under the curve.
We can use a standard normal distribution table or a calculator with a normal distribution function to determine the area under the curve of the standard normal distribution between z = -3.5 and z = 3.5. The difference between the area to the left of z = 3.5 and the area to the left of z = -3.5 is the area under the curve between these two numbers.
We determine the region to the left of z = 3.5 and z = -3.5 using a typical normal distribution table. Left of z = 3.5 is an area of 0.9998, whereas left of z = -3.5 is an area of 0.0002. Hence, the region between z = -3.5 and z = 3.5 is as follows:
0.9998 - 0.0002 = 0.9996
To turn this to a percentage, we multiply by 100:
0.9996 x 100 = 99.96%
In other words, 99.96% of the area is within 3.5 standard deviations of the mean, or between z = -3.5 and z = 3.5. This shows that most values in a normal distribution fall within a few standard deviations of the mean and accounts for a sizeable portion of the overall area under the curve.
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The figures shown are similar. What is the measure of side DE? 2 trapezoids. First trapezoid has points B, C, D, E. Distance from B to E (4.5 centimeters); B to C (6 centimeters); C to D (4.5 centimeters). Second trapezoid has points F, G, H, I. Distance from F to G (4 centimeters); G to H (3 centimeters); H to I (2 centimeters); F to I (3 centimeters). help me
The measurement of side DE is 3 centimeters.
What is the trapezoids?
A trapezoid is a 2D geometric shape that has four sides, with two sides parallel to each other and two sides non-parallel.
To solve this question, we can use the fact that similar figures have corresponding sides in proportion. Let's label the lengths of the sides of the trapezoids as follows:
First trapezoid: BE = 4.5 cm, BC = 6 cm, CD = 4.5 cm
Second trapezoid: FG = 4 cm, GH = 3 cm, HI = 2 cm, FI = 3 cm
Since the trapezoids are similar, we know that the ratio of corresponding sides is the same. Let's call this ratio k.
For the first trapezoid, the parallel sides are DE and BC, so we have:
k = DE / BC
For the second trapezoid, the parallel sides are FG and HI, so we have:
k = HI / FG
Since the trapezoids are similar, these ratios must be equal:
DE / BC = HI / FG
Substituting in the given lengths, we get:
DE / 6 = 2 / 4
Simplifying, we get:
DE = 3 cm
Therefore, the measurement of side DE is 3 centimeters.
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Question 5, please help
5/10
Answer:
1.5 gallons per minute
Step-by-step explanation:
To solve this question we have to figure out how much water there was in every time scale calculated. To do this we simply have to divide both values...
12 ÷ 8 = 1.59 ÷ 6 = 1.56 ÷ 4 = 1.53 ÷ 2 = 1.5We don't need to figure out the 0 as no time has passed for water to fill up!
We can see that for each one the answer is 1.5, so our average rate would be 1.5 gallons per minute!
Hope this helps, have a lovely day! :)
The marketing team of Yummy Cookies starts a promotion plan by giving one reward points card in each packet of cookies. It is found that 75% of the packets of Yummy Cookies contain 3‐point cards and the rest contain 7‐point cards. A total of 20 points or more can be exchanged for a gift coupon. Peggy buys 4 packets of Yummy Cookies and she opens them one by one.
a Find the probability that Peggy can exchange for a gift coupon.
The probability that Peggy can exchange for a gift coupon is 36.3%.
To find the probability that Peggy can exchange for a gift coupon, we need to calculate the probability that she obtains at least 20 reward points from the 4 packets she buys.
Let X be the total number of points Peggy obtains from the 4 packets. We can model X as a binomial random variable with n = 4 and p = 0.75, since 75% of the packets contain 3-point cards.
The probability mass function of X is:
P(X = k) = (4 choose a) * 0.75ᵃ* 0.25⁴⁻ᵃ, for a = 0, 1, 2, 3, 4
To find the probability that Peggy can exchange for a gift coupon, we need to calculate P(X >= 20). However, since the number of points in each packet is discrete and limited to 3 or 7, it is not possible to obtain exactly 20 points. Therefore, we need to find P(X >= 21).
P(X >= 21) = P(X = 21) + P(X = 22) + P(X = 23) + P(X = 24) + P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28)
Using the binomial probability mass function, we get:
P(X >= 21) = (4 choose 3) * 0.75³ * 0.25 + (4 choose 2) * 0.75² * 0.25² + (4 choose 1) * 0.75 * 0.25³ + 0.25⁴
P(X >= 21) = 0.36328125
Therefore, the probability that Peggy can exchange for a gift coupon is approximately 0.363, or 36.3%.
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What type of shape is the box of Cremora
The box of Cremora is typically rectangular in shape, with six rectangular faces and twelve edges.
Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The executives hire a
statistical consultant and ask her to determine the mean shopping time, μ, of customers at the supermarkets. The consultant will collect a random sample of
shopping times at the supermarkets and use the mean of these shopping times to estimate μ. Assuming that the standard deviation of the population of
shopping times at the supermarkets is 28 minutes, what is the minimum sample size she must collect in order for her to be 95% confident that her estimate is
within 5 minutes of μ?
Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole
number that satisfies the requirements).
(If necessary, consult a list of formulas.)
Using the formula of margin of error for the confidence interval, the minimum sample size is 120
What is the minimum sample sizeWe can use the formula for the margin of error for a confidence interval:
margin of error = z* (standard deviation/√n)
where z* is the z-score for the desired confidence level.
For a 95% confidence interval, z* = 1.96. We want the margin of error to be 5, and we are given that the standard deviation of the population is 28. So we can solve for n:
5 = 1.96 * (28 / √n)
√n = 1.96 * 28 / 5
n = (1.96 * 28 / 5)²
n = 120.47
n = 120
The minimum sample size that will give a margin of error of 5 minutes with 95% confidence is 120.
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3) What is the translation rule that describes the result of the composition of (x, y) --> (x+4, y-1) and (x, y) --> (x-5, y-5)?
The composition of (x, y) --> (x+4, y-1) and (x, y) --> (x-5, y-5) is (x, y) --> ( x - 1 , y - 6).
What is Composition of transformation ?
Each transformation applied to the prior image is combined into a composition of transformations. The result of a translation is identical to a composition of reflections across parallel lines (twice the distance between the parallel lines)
Given :
1st Translation : (x, y) --> (x+4, y-1)
It means 4 units to the right and 1 unit down.
2nd Translation : (x, y) --> (x-5, y-5)
It means 5 units to the left and 5 units down.
Now, The composition will be as follows:
(x, y) --> ( x +4 -5 , y - 1 - 5)
(x, y) --> ( x - 1 , y - 6)
It means 1 unit to the left and 6 units down.
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Jill bakes an apple pie in a pie
pan that is 10" in diameter. What is
the circumference of Jill's pie?
2
Answer: the circumference of Jill's pie is 31.4 inches.
Step-by-step explanation:
The circumference (C) of a circle is given by the formula C = πd, where d is the diameter of the circle and π (pi) is a constant approximately equal to 3.14.
In this case, the diameter of Jill's pie is 10", so we can use the formula to find the circumference:
C = πd
C = 3.14 x 10
C = 31.4 inches
Therefore, the circumference of Jill's pie is 31.4 inches.
Answer:
10pi or 31.42
Step-by-step explanation:
the formula for circumference of a circle is 2pir (yes, i can't find the symbol for pi... sorry...) (and r = the radius)
since the diameter is 2*r, the radius is 5 inches.
2*5*pi = 10pi, which is about 31.42
Write (10 + √2)÷(14+ √2) in the form a+b√2/c
where a, b and c are all integers.
Answer:
[tex]\frac{69+2\sqrt{2}}{97}[/tex]
Step-by-step explanation:
(10 + √2)÷(14+ √2)= [tex]\frac{10+\sqrt{2} }{14+\sqrt{2} }[/tex]
= [tex]\frac{(10+\sqrt{2}) x (14-\sqrt{2})}{(14+\sqrt{2}) x (14-\sqrt{2}) }[/tex]
= [tex]\frac{(10+\sqrt{2}) x (14-\sqrt{2})}{14^{2}-(\sqrt{2}^{2})}[/tex]
= [tex]\frac{{(10+\sqrt{2}) x (14-\sqrt{2})}}{196-2}[/tex]
= [tex]\frac{140-10\sqrt{2}+14\sqrt{2}-2}{194}[/tex]
= [tex]\frac{138+4\sqrt{2}}{194}[/tex]
= [tex]\frac{2 x (69+2\sqrt{2})}{194}[/tex]
= [tex]\frac{69+2\sqrt{2}}{97}[/tex]
∴ the answer is [tex]\frac{69+2\sqrt{2}}{97}[/tex] in the form [tex]\frac{a+b\sqrt{2}}{c}[/tex]
(10 + √2)÷(14+ √2) can be written in the form (69 + 3√2) ÷ 97, where a = 69, b = 3, and c = 97, and all are integers.
What are integers?A number that includes zero, positive numbers, and negative numbers, is an integer. Integer's can never be a fraction, a decimal, or a percent, it should be noted.
To write (10 + √2)÷(14+ √2) in the form a+b√2/c, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is 14 - √2.
[(10 + √2) ÷ (14 + √2)] x [(14 - √2) ÷ (14 - √2)]
= [(10 + √2) x (14 - √2)] ÷ [(14 + √2) x (14 - √2)]
= [140 + 8√2 - 2√2 - 2] ÷ [196 - 2]
= (138 + 6√2) ÷ 194
= (69 + 3√2) ÷ 97
Therefore, (10 + √2)÷(14+ √2) can be written in the form (69 + 3√2) ÷ 97, where a = 69, b = 3, and c = 97, and all are integers.
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26. Complete the following proof. Given: \( \angle Q P S \cong \angle T P R \) Prove: \( \angle Q P R \cong \angle T P S \)
The proof of the congruence of the angles ∠QPS and ∠TPR is shown below
Completing the proofs of the anglesGiven that
∠QPS ≅ ∠TPR
We have the proof of the congruence of the angles ∠TPS and ∠QPR using the following two column style of proofs
Statements Reasons
a ∠QPS ≅ ∠TPR Given
b ∠QPS ≅ ∠TPR Given
c ∠QPS ≅ ∠QPR + ∠RPS Addition property
∠TPR ≅ ∠TPS + ∠RPS
d ∠TPR ≅ ∠QPR + ∠RPS Substitution
e ∠QPR + ∠RPS ≅ ∠TPS + ∠RPS Substitution
f ∠QPS ≅ ∠TPS Subtract property
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Complete question
Complete the following proof. Given: ∠QPS ≅ ∠TPR
Prove: ∠QPS ≅ ∠TPS
Statements Reasons
a ______________ ______________
b ∠QPS ≅ ∠TPR ______________
c ∠QPS ≅ ∠QPR + ∠RPS ______________
∠TPR ≅ ∠TPS + ∠RPS
d ______________ Substitution
e ______________ ______________
f ______________ ______________
Question 6 of 10 Which of the following is the correct factorization of the polynomial below? p³-125q³ A. (p-5q) (p² + 5pq+25q²) B. (p-25q) (p2 + 25pq+25q²) C. (p² +10g)(p³ +25pq+5q²) D. The polynomial is irreducible.
Answer:
the answer is A
Step-by-step explanation:
p³-125q³
(p-5q)³
but if expand A
it will give the same p³-125q³
(p-5q)(p²+5pq+25q²)
expanding will give
p3+5p²q+25pq²-5p²q-25pq²-125q³
C.L.T
p³+5p²q-5p²q+25pq²-25pq²-125q³
p³-125q³
A curve passes through the point (0,5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
If a curve passes through the point (0,5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. The equation of the curve is y = 5e^(²x).
How to find the equation?Let's assume that the equation of the curve is y = f(x), where f(x) is some unknown function.
We are given that the slope of the curve at every point P is twice the y-coordinate of P. This means that:
f'(x) = 2f(x)
where:
f'(x) is the derivative of f(x) with respect to x.
We can solve this differential equation by separating the variables and integrating:
1/f(x) df/dx = 2
Integrating both sides with respect to x, we get:
ln|f(x)| = 2x + C
where C is a constant of integration.
To find the value of C, we can use the fact that the curve passes through the point (0,5). Substituting x=0 and y=5 into the equation, we get:
ln|f(0)| = C
Since ln|f(0)| is just a constant, we can write it as another constant, say A. Therefore:
ln|f(x)| = 2x + A
Taking the exponential of both sides, we get:
|f(x)| = e^(²x+A)
Since f(x) cannot be negative (otherwise, the slope of the curve would be negative at some points), we can drop the absolute value signs:
f(x) = e^(²x+A)
To find the value of A, we can use the fact that the curve passes through the point (0,5). Substituting x=0 and y=5 into the equation, we get:
f(0) = e^A = 5
Therefore, A = ln(5). Substituting this value into the equation, we get:
f(x) = e^(²x+ln(5)) = 5e^(²x)
So the equation of the curve is y = 5e^(²x).
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For the expression
15x² + 25x² + 5x+2/
(5x +1)
which sum represents the correct form of the partial fraction decomposition?
Answer:
15×2
Step-by-step explanation:
5×+2/(5× +1) =15×2^
100 points please help!!
Answer: (5x-1)(x+4)
Step-by-step explanation:what she said
Nine less than four times a number equals fithteen
Answer:6
Step-by-step explanation:
let's write this as equation,
4x-9=15
4x=15+9=24
x=24/4=6
A group consists of men and women. people are selected to attend a conference.
a. In how many ways can people be selected from this group of ?
b. In how many ways can women be selected from the women?
c. Find the probability that the selected group will consist of all women.
a) The number of ways to choose three people from a set of eleven is given as follows: 165.
b) The number of ways to choose three women from a set of five women is given as follows: 10.
c) The probability that the group will consist of all women is given as follows: 0.0606 = 6.06%.
What is the combination formula?The number of different combinations of x objects from a set of n elements is obtained with the formula presented as follows, using factorials.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The number of ways to choose 3 people from a set of 11 is given as follows:
C(11,3) = 11!/(3! x 8!) = 165.
The number of ways to choose 3 women from a set of 5 is given as follows:
C(5,3) = 5!/(2! x 3!) = 10.
Hence the probability that the group will consist of all women is given as follows:
10/165 = 0.0606 = 6.06%.
Missing Informationa group consists of six men and five women.Three people are selected to attend a conference. a) how many ways can three people be selected from this group of eleven?
In how many ways can three women be selected from the five women?
Find the probability that selected group will consist of all women.
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Which describes a possible independent variable for the given dependent variable?
The amount of money you earn
1. Who you call most often each day
2. The number of miles you run each day
3. The number of hours you work
4. The color of your new car
Answer:
3. The number of hours you work.
Step-by-step explanation:
Independent variable means that the occurrence of the event is not dependent on another event occurring.
In this case, the independent variable is the amount of hours you work, as you can, assumingly for the question, can work any amount you would like. However, the pay would be on the basis of how much work you do, and is dependent on the amount of hours you do (assuming you are on hourly).
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A rectangular pool has dimensions 3x feet by 2x feet The deck around the pool is 5 feet wide.
The area of the pool alone in standard form is: ___
square feet
The area of the pool and deck together is: ___
square feet
Use ^ for exponent. Do not add spaces.
Step-by-step explanation:
Pool area = L x W = 3 x * 2x = 6x^2 ft^2
Pool + deck = (L+10)(W+10) = (3x+10)(2x+10) = 6x^2 + 50x + 100 ft^2
Identify the correct leading coefficient, degree, and end behavior of P(x) = 4x^(5) + 9x^(4) + 6x^(3) - x^(2) + 2x - 7.
The leading coefficients of a polynomial is the factor of the term with the greatest degree. The largest exponent of a polynomial's variable is its degree. The coefficient for the given polynomial is [tex]4[/tex], and the degree is [tex]5[/tex].
What in mathematics is a polynomial?An expression that consists of variables, constants, and exponents that is combined using arithmetic computations like addition, subtraction, multiplication, and division is referred to as a polynomial.
What distinguishes a polynomial?By noticing which formulas only contain the operations of adding, subtract, manipulation, and quasi integer exponents, the polynomials may be located.
In the given polynomial [tex]P(x) = 4x^5 + 9x^4 + 6x^3 - x^2 + 2x - 7[/tex], the highest degree of the polynomial is [tex]5[/tex], which means it is a [tex]5^{0}[/tex] polynomial.
The coefficient of the term with the highest degree, which is [tex]4x^5[/tex], is [tex]4[/tex]. Therefore, the leading coefficient is [tex]4[/tex].
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Please ASAP Help
Will mark brainlest due at 12:00
Answer:
2
Step-by-step explanation:
-3x - 12 = 40 - 8x - 21x
-3x + 8x + 21x = 40 + 12
26x = 52
x = 52/26
x = 2
Answer:
x=-14/13, or -1 1/13
Step-by-step explanation:
Find the area of the rhombus
If the length of one side of the rhombus is 6m and the radius of the rhombus as 4 m then the area of the rhombus is 24 m².
What is the area of the rhombus?
The Area of a Rhombus = A = ½ × d1 × d2,
Where d1 and d2 are the diagonals of the rhombus.
The area of a rhombus can be found using the formula:
Area = (diagonal 1 x diagonal 2) / 2
or
Area = (base x height) / 2
We are given the length of one side of the rhombus as 6 m, which means the length of the other side is also 6 m since all sides of a rhombus are congruent. We are also given the radius of the rhombus as 4 m, which is half the length of the diagonal.
Using the Pythagorean theorem, we can find the length of the other diagonal:
diagonal 2 = 2(radius) = 2(4) = 8 m
Now we can use the formula to find the area of the rhombus:
Area = (diagonal 1 x diagonal 2) / 2
Area = (6 m x 8 m) / 2
Area = 24 m²
Therefore, the area of the rhombus is 24 m².
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Which expression is equivalent to the expression 10a - 14c
The expression 10a - 14c is factorized and then found to be equivalent to 2(5a - 7c)
How to solve an equation?An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.
Given the equation:
10a - 14c
We are to solve the equation by factorizing the expression. To factorize the expression, we need to find the term that is common to both 10a and -14c. The common term is 2, hence:
10a - 14c
2(5a - 7c)
The expression 10a - 14c is equivalent to 2(5a - 7c)
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