Answers
Part b.
If we want to dilate a figure we simply multiply each x- and y-coordinate with the scale factor we want to dilate with. Therefore:
Factor of 3
[tex]3\begin{bmatrix}{2} & {4} & {1} \\ {-1} & {3} & {5} \\ {} & {} & {}\end{bmatrix}[/tex]Multiply:
[tex]\begin{bmatrix}{6} & {12} & {3} \\ {-3} & {9} & {15} \\ {} & {} & {}\end{bmatrix}[/tex]Answer:
[tex]\begin{equation*} \begin{bmatrix}{6} & {12} & {3} \\ {-3} & {9} & {15} \\ {} & {} & {}\end{bmatrix} \end{equation*}[/tex]Related Questions
Graph the equation using the slope and the y-interceptY=-8/5x-4
Answers
We got the linear equation:
[tex]y=\frac{-8}{5}x-4[/tex]The slope of this line will be -8/5 and the y-intercept will be -4.
To graph it, first find the intercept with x - axis.
We can find it if we equal the equation to zero.
[tex]\begin{gathered} \frac{-8}{5}x-4=0 \\ \\ -\frac{8}{5}x=4 \\ \\ x=-\frac{20}{8}=-\frac{10}{4}=-\frac{5}{2} \end{gathered}[/tex]Now, we got two points:
[tex](-\frac{5}{2},0),(0,-4)[/tex]To graph the function, we only join these points:
So that's the graph for the function.
7/8 × 9/8 , but the answer as a fraction
Answers
We are given the following multiplication problem.
[tex]\frac{7}{8}\times\frac{9}{8}[/tex]To perform the fractional multiplication, simply multiply the numerators and the denominators
[tex]\frac{7}{8}\times\frac{9}{8}=\frac{7\times9}{8\times8}=\frac{63}{64}[/tex]Therefore, the result of the multiplication is 63/64
Identify the segments that are skew to line EF. There are four answers.
Answers
The lines that are not intersecting and perpendicular to each other, are not parallel to each other, and are not coplanar are called skew lines.
The four lines that are skew to line EF are AB, GB, CH, and DC.
What are skew lines?The lines that are not intersecting and perpendicular to each other, are not parallel to each other, and are not coplanar are called skew lines.
We have,
A line EF on the given square figure.
We have the following lines on the given figure.
DG, CB, HA, FA, FG, GB, AB, EH, ED, DC,CH,
We see that,
FG, FA, EH, and ED are perpendicular and intersecting to line EF.
DG, CB, and HA are parallel to line EF.
So the remaining lines will be skew to line EF
i.e AB, GB, CH, and DC
Thus,
The four lines that are skew to line EF are AB, GB, CH, and DC.
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What is the standard form for yt and factored form
Answers
Given:
The leading coefficient of a polynomial is 3.
And the roots of the polynomial is -1, 1, and 2.
Required:
To write g(t) in factored form and standard form.
Explanation:
From the given data, the factored form is given by
[tex]\begin{gathered} g(t)=3(x-(-1))(x-1)(x-2) \\ =3(x+1)(x-1)(x-2) \end{gathered}[/tex]The standard form is,
[tex]\begin{gathered} g(t)=3(x+1)(x^2-2x-x+2) \\ =3(x+1)(x^2-3x+2) \\ =3(x^3-3x^2+2x+x^2-3x+2) \\ =3(x^3-2x^2-x+2) \\ =3x^3-6x^2-3x+6 \end{gathered}[/tex]Final Answer:
The factored form:
[tex]g(t)=3(x+1)(x-1)(x-2)[/tex]The standard form:
[tex]g(t)=3x^3-6x^2-3x+6[/tex]use reference angle to find the exact value of the expression, do not use a calculator sin 2(pi)/3
Answers
Given the expression below:
[tex]\sin (\frac{2\pi}{3})[/tex]To find the exact value of the expression, let us determine the quadrant of the expression. It should be noted that the value of angles compare with the quadrants is as shown below
[tex]\begin{gathered} First\text{ quadrant, the measure of reference angle in radian is } \\ 0-\frac{\pi}{2} \end{gathered}[/tex][tex]\begin{gathered} \text{second quadrant, the measure of reference angle in radian is} \\ \frac{\pi}{2}-\pi \end{gathered}[/tex][tex]\begin{gathered} \text{third quadrant, the measure of reference angle in radian is} \\ \pi-\frac{3\pi}{2} \end{gathered}[/tex][tex]\begin{gathered} \text{fourth quadrant, the measure of reference angle in radian is} \\ \frac{3\pi}{2}-2\pi \end{gathered}[/tex]It can be observed that the expression given in the question is a fraction of (pi), greater than half of (pi) but less than (pi). This means that it lies in the second quadrant.
It should be noted that sine is positive in the second quadrant
The equivalent of the expression in the first quadrant is as shown below:
[tex]\begin{gathered} \sin (\frac{2\pi}{3})=\sin (\pi-\frac{2\pi}{3}) \\ =\sin (\frac{3\pi-2\pi}{3}) \\ =\sin (\frac{\pi}{3}) \end{gathered}[/tex][tex]\begin{gathered} \text{Therefore,} \\ \sin (\frac{2\pi}{3}),in\text{ second quadrant is the same } \\ \sin (\frac{\pi}{3}),in\text{ first quadrant.} \\ \sin (\frac{\pi}{3})=\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]Hence, the exact value of the expression is √3/2
Find the average rate of change of
refer to the image please
Answers
The average rate of change of the function f(x) = -2x² - 2 from x = 2 to x = 6 is -16
How to solve an equationAn equation shows the relationship between two or more numbers and variables.
The average rate of change of a function f(x) over the interval x = a to x = b is given by:
A(x) = [f(b) - f(a)]/[b - a]
Given that function f(x) = -2x² - 2 from x = 2 to x = 6, hence:
f(2) = -2(2)² - 2 = -10
f(6) = -2(6)² - 2 = -74
The average rate of change is:
A = [f(b) - f(a)]/[b - a]
Substituting:
A = [-74 - (-10)] / [6 - 2] = -16
The average rate of change is -16
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4505 x 219 standard algorithm
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After throughfall calculating the multiplication 4505 x 219 in standard algorithm, we have come to find that product is 986595
What is standard algorithm?In elementary math, a standard algorithm or method is a particular computation technique that is typically taught for resolving particular mathematical issues. In general, these techniques include exchanging, regrouping, long division, long multiplication using a standard notation, and standard formulas for average, area, and volume.
These techniques can vary somewhat by country and time, but they typically include these techniques. Similar techniques are also available for more complex functions like square roots, but they are no longer taught in general mathematics classes because calculators are more convenient.
4505 x 219 in standard algorithm is as follows:
⇒ 4 5 0 5
× 2 1 9
4 0 5 4 5
+ 4 5 0 5
+ 9 0 1 0
9 8 6 5 9 5
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The standard height from the floor to the bull's-eye at which a standard dartboard is hung is 5 feet 8 inches. A standard dartboard is 18 inches in diameter.
Suppose a standard dartboard is hung at standard height so that the bull's-eye is 12 feet from a wall to its left.
Brian throws a dart at the dartboard that lands at a point 11.5 feet from the left wall and 5 feet above the floor.
Does Brian's dart land on the dartboard?
Answers
The equation of the circle that represents the dartboard is (x - 12)² + (y - 17/3)² = 9/16, where the origin is the lower left corner of the room and the unit of the radius is feet.
The position of Brian's dart is represented by the coordinates (11.5, 5). Brian's dart does land on the dartboard.
What is the equation of a circle?Mathematically, the standard form of the equation of a circle is represented by this mathematical expression;
(x - h)² + (y - k)² = r²
Where:
h and k represents the coordinates at the center.r represents the radius of a circle.From the question, we have the following information:
The height of this standard dartboard, k = 5 feet, 8 inches.
The diameter of this standard dartboard = 18 inches.
The bull's eye, h = 12 feet.
Next, we would convert the all of the units in inches to feet as follows:
Height, k = 5 + 8/12
Height, k = 5 + 2/3
Height, k = 17/3 feet.
For the diameter, we have:
Diameter = 18/12
Diameter = 3/2 feet.
Also, we would determine the radius as follows:
Radius, r = diameter/2
Radius, r = (3/2)/2
Radius, r = 3/4 feet.
Substituting the parameters into the standard equation, we have;
(x - 12)² + (y - 17/3)² = (3/4)²
(x - 12)² + (y - 17/3)² = 9/16
Next, we would determine whether Brian's dart land on the dartboard:
(x - 12)² + (y - 17/3)² < 9/16
(x - 12)² + (y - 17/3)² < 9/16
(11.5 - 12)² + (5.5 - 5.67)² < 0.5625
0.25 + 0.0289 < 0.5625
0.2789 < 0.5625 (Yes, it does land because it's within the circumference of this standard dartboard).
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If the formular-(X-X|Y-Xwere used to find the r-value of the5x буfollowing data, what would be the value of x?XY8496101811912|13A. 8B. 10C. 6D. 4
Answers
The value of x⁻, is the mean of the x values of the table.
The mean of x is obtained by adding all values of x, and divided the result by the total number of data, which is 5.
Then, you have:
[tex]\bar{x}=\frac{8+9+10+11+12}{5}=\frac{50}{5}=10[/tex]Hence, the mean of x, which is used in the formula to calculate the r-value, is 10
Express your answer in scientific notation.
5.4x10^5 + 6.7x10^4
Answers
Answer:
Step-by-step explanation:
5.4*10*10*10*10*10+6.7* 10*10*10*10
540,000+6.7*10,000
540,000+60,000
114,000=1.14*10^5
What is the slope of the line that contains these points? 9 X 17 13 21 -24 -21 -18 Y - -15 slope:
Answers
We have to use the slope formula
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Let's replace the points (9,-24) and (13,-21). Where
[tex]\begin{gathered} x_1=9 \\ x_2=13 \\ y_1=-24 \\ y_2=-21 \end{gathered}[/tex][tex]m=\frac{-21-(-24)}{13-9}=\frac{-21+24}{4}=\frac{3}{4}[/tex]Hence, the slope is 3/4.At 30 mph a trip from Town A to Town B takes 6 hours. If the speed were doubled, how many hours would the trip take?
Answers
The time taken by Town A to Town B will be 3 hours.
What is speed?Speed is defined as the ratio of the time distance traveled by the body to the time taken by the body to cover the distance. Speed is the ratio of the distance traveled by time. The unit of speed in miles per hour.
Given that at 30 mph a trip from Town A to Town B takes 6 hours.
The time taken will be calculated as,
Total distance = 30 x 6 = 180 meters
Speed = 30 x 2 = 60 mph
The time,
T = 180 / 60= 3 hours
Therefore, the time taken by Town A to Town B will be 3 hours.
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please help me brainliest?
Answers
Answer:
See below.
Step-by-step explanation:
Given multiplication:
[tex]3 \dfrac{1}{5} \times 2 \dfrac{5}{8}[/tex]
Convert the mixed numbers into improper fractions by multiplying the whole number by the denominator of the fraction, adding this to the numerator of the fraction, and placing the answer over the denominator:
[tex]\implies \dfrac{3 \times 5+1}{5} \times \dfrac{2 \times 8+5}{8}[/tex]
[tex]\implies \dfrac{15+1}{5} \times \dfrac{16+5}{8}[/tex]
[tex]\implies \dfrac{16}{5} \times \dfrac{21}{8}[/tex]
[tex]\textsf{Apply\;the\;fraction\;rule} \quad \dfrac{a}{b} \times \dfrac{c}{d}=\dfrac{ac}{bd}:[/tex]
[tex]\implies \dfrac{16 \times 21}{5 \times 8}[/tex]
[tex]\implies \dfrac{336}{40}[/tex]
Reduce the fraction by dividing the numerator and denominator by the common factor of 8:
[tex]\implies \dfrac{336 \div 8}{40 \div 8}[/tex]
[tex]\implies \dfrac{42}{5}[/tex]
Divide the numerator by the denominator:
[tex]\implies 42 \div 5=8\;\textsf{remainder}\;2[/tex]
The mixed number answer is the whole number and the remainder divided by the denominator:
[tex]\implies 8 \dfrac{2}{5}[/tex]
Step-by-step explanation:
when you want to start solving it you will do 16 over 5 * 21/8equals 8 whole number 2/5
it will give you 336 all over 40 if you divide by 8 you will get 42/5 which is 8 whole number 2/5
Choose all equivalent expression ( s). (4) ^ (3z ^ 2); (4) ^ (- 3x ^ 2); (1/4) ^ (3z ^ 2); (pi/4) ^ (3x)
Answers
Answer:
B, C
Step-by-step explanation:
You want to find the equivalent expressions among ...
4^(3x^2)4^(-3x^2)(1/4)^(3x^2)(x/4)^(3x)Rules of exponentsThe relevant rule of exponents is ...
a^b = (1/a)^-b
Equivalent expressionsExpressions with the same base and different exponents will not be equivalent. (A≠B).
Expressions with different bases and the same exponent will not be equivalent. (A≠C).
A variable in the base cannot replace a variable in the exponent. (A≠D).
The rule of exponents tells us replacing the base by its reciprocal and negating the exponent will result in an equivalent expression. (B=C).
The equivalent expressions are 4^(-3x^2) and (1/4)^(3x^2).
Poland Spring Hotel is a 500-room property that offers only rooms, no F&B service. You are in the process of evaluating the business as an investment. Calculate the breakeven sales and rooms using the information below
Answers
Answer: GAS
Step-by-step explanation:
A security keypad uses five digits (0 to 9) in a specific order. How many different
keypad patterns are possible if the first three digits must be even and the last digit
cannot be zero?
Answers
The different keypad patterns that are possible if the first three digits must be even and the last digit cannot be zero is 17500 ways.
How to calculate the value?It should be noted that the security keypad uses five digits (0 to 9) in a specific order. On this case, the numbers from 0 to 9 make up 10 numbers.
In this case, there are 5 even numbers.
The total number of possible codes will be:
= 5 × 5 × 10 × 10 × 7
= 17500
There are 17500 ways.
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What is the polynomial function of lowest degree with rational real coefficients and roots 2 and squared root 5?
Answers
The polynomial having the above quantities is expressed as x³ - 2x² - 5x + 10 = 0
The above situations will form a Cubic polynomial. A Cubic polynomial function may be defined as an expression which can be written in the form of ax³ + bx² + cx + d = 0 where a, b, c and d are coefficients and x is the independent variable. Since, the polynomial should have real coefficients so its multiplicity will be equal to 1. Now, the roots of the polynomial are given as 2 and √5. Since, square roots always occur in pair so the polynomial will have -√5 as its root. Now, the polynomial formed will be
(x - 2) (x - √5) (x + √5) = 0
(x - 2) (x² - 5) = 0
x³ - 5x - 2x² + 10 = 0
=> x³ - 2x² - 5x + 10 = 0 which is the required polynomial.
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Given lines l, m and n are all parallel and cut by two transversal lines, find the value of 2. 2 12 6. т х 33 n
Answers
Answer:
66
Explanation:
By the Thales theorem, if we have the following figure:
Then, the following equation applies:
[tex]\frac{a}{c}=\frac{b}{d}[/tex]So, in this case, we can formulate the following equation:
[tex]\frac{6}{12}=\frac{33}{x}[/tex]Then, solving for x, we get:
[tex]\begin{gathered} 6\cdot x=33\cdot12 \\ 6x=396 \\ \frac{6x}{6}=\frac{396}{6} \\ x=66 \end{gathered}[/tex]Therefore, the value of x is 66.
12Solve using the Quadratic Formula for 2x2 + 5x – 3 = 0x = -5, 7b. X = -12,2X = -3,42
Answers
2x² + 5x - 3 = 0
Multiply the coeeficient of x² and the constant (-3)
That is 2 ( - 3) = -6
Find the numbers whose sum is 5 and whose product is -6
The number is 6 and -1
Replace 5x by the numbers
2x² + 6x - x - 3 = 0
2x( x + 3) - 1 ( x + 3) = 0
(2x - 1 ) ( x + 3) = 0
Either 2x -1 = 0 or x + 3 = 0
2x = 1
x = 1/2 or x = -3
A print shop borrows $4200 from a credit union for 178 days. The credit union charges simple interest at an annual rate of 6.5% for this loan. Assume each day is 1/365 of a year.
round your final answers to the nearest cent.
A) Find the interest that will be owed after 178 days.
B) Assuming the print shop doesn't make any payments, find the amount owed after 178 days.
Answers
Using percentages, the answer to both the subparts are:
(A) Interest that will be owed after 178 days: $4,336.5(B) Interest after 178 days if sop doesn't repay: $4,473What is the percentage?A percentage is a figure or ratio stated as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and occasionally "pc" is also used, the percent symbol, "%," is frequently used to indicate it. A % is a number without dimensions and without a standard measurement.So, since the interest is annual 6.5%:
(A) Interest that will be owed after 178 days:
Interest/2 ⇒ 6.5/2 ⇒ 3.25%Now, 3.25% of 4200:
4200/100 × 3.25 = 136.5$4,336.5(B) Interest after 178 days if sop doesn't repay:
Interest ⇒ 6.5%Now, 6.5% of 4200:
4200/100 × 6.5 = 273$4,473Therefore, using percentages, the answer to both the subparts are:
(A) Interest that will be owed after 178 days: $4,336.5(B) Interest after 178 days if sop doesn't repay: $4,473Know more about percentages here:
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Question #36 Multiple-Choice Reduce -2 + b2 by 7 + b2. C5 0 2b2-7 C9
Answers
Answer:
[tex]-2+b^2-7-b^2=-9[/tex]Step-by-step explanation:
Reduce the following expressions:
[tex]-2+b^2-(7+b^2)[/tex]Operate like terms:
[tex]-2+b^2-7-b^2=-9[/tex]The length of the smaller rectangle is 12 inches and the width is x inches. The length of the larger rectangle is 20 inches and the width is 15 inches. What is the area of the smaller rectangle?
Answers
The area of the smaller rectangle is 108 inches².
What is the Area of a Rectangle?The area of a rectangle is calculated to determine the area occupied by the rectangle within its boundary. A rectangle's area (A) is the product of its length 'a' and width or breadth 'b'. As a result, the Area of the Rectangle = (a × b) square units.
Given the dimensions of the smaller rectangle:
Length = 12 inches
Width = x inches
Given the dimensions of the larger rectangle:
Length = 20 inches
Width = 15 inches
To find the value of x we use proportion,
[tex]\frac{12}{x} = \frac{20}{15}[/tex]
Cross-multiplying, we get
12 × 15 = 20x
x = 180/20 = 9
Hence, the width of the smaller rectangle is 9 inches.
Area of the smaller rectangle = length × width
= 12 × 9
= 108 inches²
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harry and marie despoit $800.00 into a savings account which earns 9% interest compounded monthly they want to use the money in the account to go on a trip in 3 years how much will they be able to spend
Answers
harry and marie despoit $800.00 into a savings account which earns 9% interest compounded monthly they want to use the money in the account to go on a trip in 2 years how much will they be able to spend
we know that
The compound interest formula is equal to
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest in decimal
t is Number of Time Periods
n is the number of times interest is compounded per year
in this problem we have
P=$800
r=9%=9/100=0.09
n=12
t=2 years
substitute in the expression above
[tex]\begin{gathered} A=800(1+\frac{0.09}{12})^{12\cdot2} \\ \\ A=800(\frac{12.09}{12})^{(24)} \\ A=\$957.13 \end{gathered}[/tex]the answer is $957.13The list price of a DVD player is $129. Sam found the player on sale for $90.30. What was the percent discount?
Answers
A percent discount of 30%
Explanation:
Since $129 represent 100% of the price, then the new price $90.30 represent 70 %
129 * 100 / 90.30 = 70 %
Since from the 100% of the price, it went to 70% of it, there was a discount of 30%
100 - 70 = 30%
A local farm raises horses and goats. • Let x represent the number of horses on the farm. • The number of goats is 3 times the number of horses. • The total number of horses and goats on the farm is 96. What is x, the number of horses on the farm?
Answers
The number of horses on the harm will be 24.
What is an expression?
Expression in math is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Given that;
A local farm raises horses and goats.
The total number of horses and goats on the farm is 96.
Let number of horses = x
Since, The number of goats is 3 times the number of horses.
Then, Number of goats = 3x
The total number of horses and goats on the farm is 96.
Hence, we can formulate;
x + 3x = 96
4x = 96
x = 24
Thus, The number of horses on the harm will be 24.
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Instructions: Solve the system. Enter your answer as an ordered pair.
Answers
Given the system:
[tex]\begin{gathered} 3x-9y=12\text{ Eq. 1} \\ 3x+4y=-1\text{ Eq. 2} \end{gathered}[/tex]First, we solve for 3x on both equations, as follows:
[tex]\begin{gathered} 3x=12+9y \\ 3x=-1-4y \end{gathered}[/tex]Equating both equations:
[tex]\begin{gathered} 12+9y=-1-4y \\ 13=-4-9y \\ 13=-13y \\ y=-1 \end{gathered}[/tex]Substituting y on equation 2:
[tex]\begin{gathered} 3x+4y=-1\text{ Eq. 2.} \\ 3x+4\times(-1)=-1 \\ 3x=-1+4 \\ 3x=3 \\ x=1 \end{gathered}[/tex]ANSWER
(1, -1)
Shona is bundling magazines to recycle he notices at 6 magazines weigh 5/8 pound in all and that the magazines all weigh the same amount. What is the unit rate for pounds per magazine?I don't understand this at all
Answers
Given data:
The given weight of 6 magazines is 5/8 pound.
The given expression is,
6M=5/8 pounds
6M=0.625 pounds
1M=0.1041667 pounds
The weight of one magzine is 0.104167 pounds.
Maya has 846 beads. She is making bracelets with 18 beads on each. How many bracelets is she able to make?
Answers
Answer:
maya is able to make 47 bracelets
Step-by-step explanation:
Given the function: f(x) = (x − 3)(x + 7)(x − 1)its f-intercept is:its x-intercepts are:
Answers
Given: Given the function: f(x) = (x − 3)(x + 7)(x − 1)
Required: its f-intercept and its x-intercepts.
Explanation:
[tex]f(x)=(x-3)(x+7)(x-1)[/tex]For f-intercept, put x = 0
[tex]\begin{gathered} f(0)=(0-3)(0+7)(0-1) \\ f(0)=21 \end{gathered}[/tex]So the f-intercept is (0,21).
For x-intercepts, put f(x) = 0
[tex]\begin{gathered} 0=(x-3)(x+7)(x-1) \\ x=3,-7,1 \end{gathered}[/tex]So x-intercepts are (3,0),(-7,0) and (1,0).
Final Answer:
f-intercept: (0,21)
x-intercept: (3,0),(-7,0),(1,0).
Compute the horizontal force P required to prevent the block from sliding down the plane for the 100 lb block shown. Assume the coefficient of static friction to be 0.65.
Answers
canThe first First step we need to do is to make the decomposition of the vectors W and P.
Both will have a component perpendicular and parallel to the plane. The perpendicular will be used to calculate the maximum static friction force, and the horizontal will be used to find the P required to prevent the block from sliding.
From the sketch, we are able to define both, parallel and perpendicular components of P, as it follows:
[tex]\begin{gathered} P_{\text{//}}=P\cos (30\degree)=\frac{P\sqrt[]{3}}{2} \\ P_{perp}=P\sin (30\degree)=\frac{P}{2} \end{gathered}[/tex]Now, we can do the same for W.
And for W we also can provide the two components as follows:
[tex]\begin{gathered} W_{//}=W\sin (30\degree)=\frac{W}{2} \\ W_{\text{perp}}=W\cos (30\degree)=\frac{W\sqrt[]{3}}{2} \end{gathered}[/tex]Now, we can elaborate on both equations: one for the perpendicular direction and the other for parallel. In the perpendicular direction, we have a component of W, one component of P, and the normal force N. Because the block is going to move, or change its movement along this direction, the sum of the forces pointing upwards must be equal to the sum of the forces pointing downwards. From this, we can write the following:
[tex]\begin{gathered} N=P_{\text{perp}}+W_{\text{perp}} \\ N=\frac{P}{2}+\frac{W\sqrt[]{3}}{2}=\frac{P+W\sqrt[]{3}}{2} \end{gathered}[/tex]Now, for the horizontal, we have the P component to the right and the W component to the left. If we imagine the block is almost sliding. We can write the following equation, from the premise the forces will cancel each other just like the perpendicular case:
[tex]\begin{gathered} P_{//}+F_{\mu}=W_{//} \\ \frac{P\sqrt[]{3}}{2}+N\times\mu_{static}=\frac{W}{2} \end{gathered}[/tex]Here it is used the fact that the friction force is equal to the multiplication of the coefficient of static friction by the normal force. Here we assumed also that the friction is maximum because the block is on the verge of motion downwards, and for this reason, the Friction is upwards, with the P component.
Now, substituting N and the coefficient, we find:
[tex]\begin{gathered} \frac{P\sqrt[]{3}}{2}+\frac{P+100\sqrt[]{3}}{2}0.65=\frac{100}{2}=50 \\ \frac{P(\sqrt[]{3}+0.65)+65\sqrt[]{3}}{2}=50 \\ P(\sqrt[]{3}+0.65)+65\sqrt[]{3}=100 \\ P(\sqrt[]{3}+0.65)=100-65\sqrt[]{3} \\ P=\frac{100-65\sqrt[]{3}}{\sqrt[]{3}+0.65}\cong\frac{100-65\times1.732}{1.732+0.65}=\frac{100-112.58}{2.382} \\ P=-\frac{12.58}{2.382}\cong-5.275\text{lbf} \end{gathered}[/tex]From this, we can see that the force P made to the left with an intensity equal to -5.275 lbf will bring the block on the verge of motion downwards. If we consider that P is strong enough to make it almost move upwards, it is, the Normal Force will be downwards, we can remake the calculation as it follows:
[tex]\begin{gathered} P_{//}=W_{//}+F_{\mu} \\ \frac{P\sqrt[]{3}}{2}=\frac{W}{2}+N\times\mu_{static} \end{gathered}[/tex]And substituting values, we have:
[tex]\begin{gathered} \frac{P\sqrt[]{3}}{2}=50+\frac{P+100\sqrt[]{3}}{2}0.65 \\ \frac{P(\sqrt[]{3}-0.65)}{2}=50+50\sqrt[]{3}\times0.65 \\ P=\frac{2}{\sqrt[]{3}-0.65}\times50(1+\sqrt[]{3}\times0.65)\cong196.46 \end{gathered}[/tex]From this, we know that the max value for P, where the block will not slide is going to be 196.46 lbf to the right.