Since all the sides of the figure have the same length, then the figure is a rhombus. Then, its diagonals intersect at an angle of 90°.
Let O be the intersection of the diagonals of the rhombus. Notice that the triangle EOA is a right triangle. Since the side EA is the hypotenuse of the triangle, then, recalling the trigonometric functions:
[tex]\begin{gathered} \cos (30)=\frac{EO}{EA} \\ \sin (30)=\frac{OA}{EA} \end{gathered}[/tex]Use this information to solve for the segments EO and OA:
[tex]\begin{gathered} EO=EA\cdot\cos (30) \\ =10\cdot\frac{\sqrt[]{3}}{2} \\ =5\cdot\sqrt[]{3} \end{gathered}[/tex][tex]\begin{gathered} OA=EA\cdot\sin (30) \\ =10\cdot\frac{1}{2} \\ =5 \end{gathered}[/tex]Since the diagonal EM is twice the segment EO and the diagonal BA is twice the segment OA, then the lengths of the diagonals are:
[tex]\begin{gathered} BA=10 \\ EM=10\cdot\sqrt[]{3} \end{gathered}[/tex]Therefore, the answer is:
[tex]10\text{ yards and }10\cdot\sqrt[]{3}\text{ yards}[/tex]Fnd the volume of each cylinder below.9.18 in15 in
9) We can calculate the volume as the product of the base area and the height.
The base is a circle with radius r=18 in. Then, its area is:
[tex]A_b=\pi r^2=\pi\cdot18^2=324\pi[/tex]Then, we can calculate the volume V as:
[tex]V=A_b\cdot h=324\pi\cdot15=4860\pi[/tex]10) In this case the circular base is on the side, but we can still use the same principle to calculate the volume.
The area of the base with diameter D = 11 in is:
[tex]A_b=\frac{\pi D^2}{4}=\frac{\pi\cdot11^2}{4}=\frac{\pi\cdot121}{4}=\frac{121}{4}\pi[/tex]Then, we can calculate the volume V as:
[tex]V=A_b\cdot h=\frac{121}{4}\pi\cdot21=\frac{2541}{4}\pi=635.25\pi[/tex]Answer:
9) V = 4860π
10) V = 635.25π
Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 0 to 10, with 10 being the happiest. A line was fit to the data to model the relationship.Which of these linear equations best describes the given model?A) ŷ = 5x + 1.5B) ŷ = 1.5x + 5Or C) ŷ = -1.5x + 5Based on this equation, estimate the mood rating for a student that spent 2.5 hours playing sports.Round your answer to the nearest hundredth.__________.
We have to relate a linear function (the regression model) with its equation.
We can see in the graph that the y-intercept, the value of y(0), is b=5.
Then, we can estimate the slope with the known points (0,5) and (2,8):
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{8-5}{2-0}=\frac{3}{2}=1.5[/tex]Then, with slope m=1.5 and b=5, the regression model equation should be:
[tex]y=1.5x+5[/tex]We can estimate the mood for students that spent 2.5 hours playing sports by replacing x with 2.5 in the model and calculate y:
[tex]y(2.5)=1.5\cdot2.5+5=3.75+5=8.75[/tex]NOTE: we could also have look on the graph instead of doing the calculation.
Answer: B) y=1.5x+5
The estimation of the mood for a student that spent 2.5 hours playing sports is 8.75.
5) Each table represents a proportional relationship. (From Unit 2 Lesson 2) a) Fill in the missing parts of the table. b) Draw a circle around the constant of proportionality. a х у a b т n 2 10 12 3 15 20 10 3 735 5 10 18 1 1 1
Given:
The table represents a proportional relationship.
a) To find the missing values of table,
For first table,
[tex]\begin{gathered} \frac{10}{2}=\frac{15}{x} \\ 10x=15\times2 \\ 10x=30 \\ x=\frac{30}{10}=3 \\ \frac{15}{3}=\frac{y}{7} \\ 15\times7=3y \\ 3y=105 \\ y=\frac{105}{3}=35 \\ \frac{35}{7}=\frac{y}{1} \\ 35=7y \\ y=\frac{35}{7}=5 \end{gathered}[/tex]For second table,
[tex]\frac{3}{12}=\frac{b}{20}=\frac{10}{a}=\frac{b}{1}[/tex][tex]\begin{gathered} \frac{3}{12}=\frac{b}{20} \\ 3\times20=12y \\ b=\frac{60}{12}=5 \\ \frac{5}{20}=\frac{10}{a} \\ 5a=10\times20 \\ a=\frac{200}{5}=40 \\ \frac{10}{40}=\frac{b}{1} \\ 40b=10 \\ b=\frac{10}{40}=\frac{1}{4} \end{gathered}[/tex]For third table,
[tex]\begin{gathered} \frac{3}{5}=\frac{n}{10}=\frac{18}{m}=\frac{n}{1} \\ \frac{3}{5}=\frac{n}{10} \\ 30=5n \\ n=\frac{30}{5}=6 \\ \frac{3}{5}=\frac{18}{m} \\ 3m=90 \\ m=30 \\ \frac{3}{5}=\frac{n}{1} \\ 5n=3 \\ n=\frac{3}{5} \end{gathered}[/tex]b) To draw the circle around the constant of proportionality.
For first table the constant of proportionality is 5.
For second table the constant of proportionality is 1/4.
For third table the constant of proportionality is 3/5 .
Every week Ben collects a few pounds of paper to recycle. The graph below shows the total number of pounds of paper(y) that Ben collected in a certain amount of time (x), in weeks:
To obtain the amount of paper that would most likely be collected in 10 weeks, the following steps are necessary:
Step 1: Select two points that lie on the straight line and use the two points to derive the equation of the straight line, as follows:
Such two points could be: (x1, y1) = (0, 30) and (x2, y2) = (120, 3)
Using the following formula, we can derive the equation of the straight line:
[tex]\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}[/tex]Thus:
[tex]\begin{gathered} \frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1} \\ \Rightarrow\frac{y-30_{}}{x-0_{}}=\frac{120_{}-30_{}}{3_{}-0_{}} \\ \Rightarrow\frac{y-30_{}}{x_{}}=\frac{90_{}}{3_{}_{}} \\ \Rightarrow\frac{y-30_{}}{x_{}}=30 \\ \Rightarrow y-30=30\times x \\ \Rightarrow y=30x+30 \end{gathered}[/tex]The above equation can be re-written as:
[tex]\begin{gathered} y=30x+30 \\ \Rightarrow\text{Amount of paper collected = 30 }\times number\text{ of w}eeks\text{ + 30 } \end{gathered}[/tex]Step 2: Use the derived equation to obtain the value of the amount of paper collected in 10 weeks, as follows:
In 10 weeks, we will have :
[tex]\begin{gathered} \text{Amount of paper collected = 30 }\times number\text{ of w}eeks\text{ + 30 } \\ \Rightarrow\text{Amount of paper collected = 30 }\times10\text{ + 30 } \\ \Rightarrow\text{Amount of paper collected = 300 + 30 }=330 \\ \Rightarrow\text{Amount of paper collected = 3}30 \end{gathered}[/tex]Therefore, the amount of paper that would most likely be collected in 10 weeks is 330
at 37 ft string of lights will be attached to the top of a 35 ft pole for Holiday display how far from the base of a pool should the end of the string of lights be anchored
Answer:
12 feet
Explanation:
The diagram representing the problem is attached below:
The distance of the pole to the base of the string is the value x.
Using Pythagoras Theorem:
[tex]\begin{gathered} 37^2=35^2+x^2 \\ x^2=37^2-35^2 \\ x^2=144 \\ x^2=12^2 \\ x=12ft \end{gathered}[/tex]The end of the string should be anchored 12 ft from the base of the pole.
Select the correct choice and fill in the answer box
To begin with, let us look at a few definitions that will help
A relation is a function if each x-value is paired with exactly one y-value. A vertical line test on a graph can be used to determine whether a relation is a function.
If we use a graph to check, we will have
We can see that there is no overlapping of coordinates. The table satisfies the vertical line test.
Hence, it is a function
The domain and range of function is the set of all possible inputs and outputs of a function respectively. The domain and range of a function y = f(x) is given as domain= {x ,x∈R }, range= {f(x), x∈Domain}.
The domain of the function, D is given by
[tex]D=\mleft\lbrace-1,0,1,2,3\mright\rbrace[/tex]The range, R is given by
[tex]R=\mleft\lbrace-6,-1,2,5,8\mright\rbrace[/tex]m/cour16121/quizzes/2919544/take22A totem pole casts a shadow 45 feet long at the same time that a 6-foot-tallmancasts a shadow that is 3 feet long. Find the height of the totem pole.Height of the totemfeetHint: set a proportion like this oneshadow totemshadow manactual height totem actual height manucation
The totem pole with its shadow form a right triangle, and the same happens with the man and its shadow. Since it happens at the same time of the day, the "inclination" of the sunlight is the same for both, and those two triangles are similar, which means that their corresponding ratiso betweens sides are the same.
Using the hint given on the problem, we can estabilish the following equation for the height of the totem(let's call it h).
[tex]\frac{45}{h}=\frac{3}{6}[/tex]Now, we just need to solve for h.
[tex]\begin{gathered} \frac{45}{h}=\frac{3}{6} \\ \frac{45}{h}=\frac{1}{2} \\ \frac{h}{45}=2 \\ h=2\cdot45 \\ h=90 \end{gathered}[/tex]The height of the totem pole is 90 ft.
ХI went to the bank 4 times last week and withdrew a total of $160. What was theaverage amount withdrew each time? Write and solve an expression to find theanswer
Sound is measured in decibels, using the formula d = 10 log() where P is the intensity of the sound and P, is the weakest sound the human ear can hear. A horn has a decibel warning of20. How many times more intense is this horn compared to the weakest sound heard to the human ear?
We have the following:
The formula is the following
[tex]d=10\log (\frac{P}{P_o})[/tex]From what the statement tells us, the value of d is equal to 20, now we replace and solving for P / Po
[tex]\begin{gathered} 10\log (\frac{P}{P_o})=20 \\ \frac{10}{10}\log (\frac{P}{P_o})=\frac{20}{10} \\ \log (\frac{P}{P_o})=2\rightarrow\log (x)=2\rightarrow x=10^2\rightarrow x=100 \\ \frac{P}{P_o}=100 \\ P=100\cdot P_o \end{gathered}[/tex]Therefore, the sound of the horn is about 100 more intense than the weakest sound heard to the human ear.
Which one of the following graphs represents the solution of the inequality 2x + 1 ≥ 3?A.-3-2-1 0 123B.++-3-2-1 0123-3-2-1 0 123-3-2-1 0 1 2 3OC.OD.
The inequality given is:
[tex]2x+1\ge3[/tex]Let's solve for x:
[tex]\begin{gathered} 2x\ge3-1 \\ . \\ x\ge\frac{2}{2} \\ . \\ x\ge1 \end{gathered}[/tex]The solution set of this inequality is the set of all numbers bigger or equal than 1.
Thus, the correct answer is option A, where we can see that the values start at 1 and goes to positive infinity.
Simplify (sqrt)98m^12 using factor tree or splitting up using perfect squares. Quick answer showing work = amazing review :)
The expression is:
[tex]\sqrt[]{98m^{12}}[/tex]We need to use a factor tree to solve the problem. We will draw it as shown below:
According to the factor tree we can represent the 98 as:
[tex]\sqrt[]{2\cdot7^2m^{12}}[/tex]We can now remove the terms that have a power of 2 and a power of 12. For that we need to divide the exponents by 2.
[tex]7^{}\sqrt[]{2}m^6[/tex]The simplified expression is 7*sqrt(2)*m^6.
DreviousRandom numbers are useful forA creatingOB. beingOC. modelingOD. sellingReset Selectionreal-world situations that involve chance.
Given random numbers, it is important to remember that Random Numbers are defined as those numbers that each have the same probability of being selected.
In Statistics, random numbers are useful to model different real-world situations.
An example of random numbers is the numbers of a lottery.
Another example of random numbers is the numbers obtained by rolling a numbered dice.
Hence, the answer is: Option C.
The Rainforest Pyramid at Moody Gardens is a rectangular pyramid that has glass sides to allow the sunlight to enter the building. The base of the building is approximately 320 feet wide and 200 feet long. The height of each side is approximately 140 feet. How many square feet of glass were needed to cover the lateral sides of the pyramid?
We have the following diagram for a rectangular pyramid:
Where:
• h is the height of the pyramid,
,• s is the slant height, the height of the sides.
The area of the sides (without the base) of the pyramid is given by:
[tex]A=\frac{1}{2}\cdot p\cdot s\text{.}[/tex]Where p is the perimeter of the base.
We have a rectangular pyramid with:
• base with sides a = 320 ft and b = 200 ft,
,• slant height (heigh of the sides ) s = 140 ft.
The perimeter of the base is:
[tex]p=2\cdot(a+b)=2\cdot(320ft+200ft)=1040ft\text{.}[/tex]Replacing the value of p and s in the formula of the area we get:
[tex]A=\frac{1}{2}\cdot(1040ft)\cdot140ft=72800ft^2.[/tex]Answer
It was needed 72800 ft² of glass.
01 Question 11 What is the area of the shaded region? 12cm 6 cm 10cm 8cm 128cm? 96cm2 X 144cm? a 112cm?
We can calculate the area of the shaded region as the difference between the area of the rectangle (sides 12 cm and 10 cm) and the area of the triangle in the corner.
The triangle has base of b=12-8=4 cm and height h=10-6=4 cm.
Then, we can calculate the area of the shaded area as:
[tex]\begin{gathered} A=A_r-A_t \\ A=b_r\cdot h_r-\frac{b_t\cdot h_t}{2} \\ A=12\cdot10-\frac{4\cdot4}{2} \\ A=120-\frac{16}{2} \\ A=120-8 \\ A=112\operatorname{cm}^2 \end{gathered}[/tex]Answer: 112 cm^2
1) Find the angle in degrees without using a calculator: a) arcsin( √3/2)
Then, since arcsin is a function:
[tex]\begin{gathered} R\rightarrow\mleft\lbrace-1;\text{ 1}\mright\rbrace \\ We\text{ take only value }\theta=\frac{\pi}{3},\text{ without the periodic values.} \\ \text{That means,} \\ \arcsin (\frac{\sqrt[]{3}}{2})=60\text{ degrees= }\frac{\pi}{3} \end{gathered}[/tex]given the following trig equation, find the Exact value of the remaining 5 trig functionstan (theta) = 5/6 and cos theta < 0Start by drawing the triangle in standard position and use the Pythagorean theorem to find the remaining side. A. label the exact value of all 3 sides of the triangle drawn in the correct quadrantB. DETERMINE the EXACT value of the remaining 5 trig functions! (sin) (cos) (tan) (sec) (csc) (cot)
tan (theta) = 5/6 and cos theta < 0
tan (theta) = 5/6 ==> theta = tan^-1(5/6) = 39.80557109
theta = 39.80557109
cos(theta) = cos(39.80557109) = 0.7682212796
It says that cos(theta) < 0, so the 39.80557109 degrees is in rality an angle of 90 + 39.80557109 = 120.80557109
sin (theta) = sin (120.80557109) = 0.858910105
cos(theta) = cos(120.80557109) = −0.5121263824
tan(theta) = tan(120.80557109) = −1.677144811
sec(theta) = sec(120.80557109) = −1.952643008
csc(theta) = csc(120.80557109) = 1.164266195
cot(theta) = cot(120.80557109) = −0.5962514348
which of the following represents a line that is parallel to the line with equation y = – 3x + 4 ?A. 6x +2y = 15B. 3x - y = 7 C. 2x - 3y = 6D. x + 3y = 1
In order to have parallel lines, the slopes of the lines need to be the same.
In order to check the slope for each option, let's put the equations in the slope-intercept form:
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
So we have that:
A.
[tex]\begin{gathered} 6x+2y=15 \\ 2y=-6x+15 \\ y=-3x+7.5 \end{gathered}[/tex]B.
[tex]\begin{gathered} 3x-y=7 \\ y=3x-7 \end{gathered}[/tex]C.
[tex]\begin{gathered} 2x-3y=6 \\ 3y=2x-6 \\ y=\frac{2}{3}x-2 \end{gathered}[/tex]D.
[tex]\begin{gathered} x+3y=1 \\ 3y=-x+1 \\ y=-\frac{1}{3}x+\frac{1}{3} \end{gathered}[/tex]So the only option with a line with a slope of -3 is Option A.
10. Linek is graphed below. Write an equation for line m that is perpendicular to line (there are multiple correct answers).
step 1
Find the equation of line k
the slope of line k is
m=-3/4 ------> previous answer
step 2
If two lines are perpendicular, then their slopes are opposite reciprocal
so
the slope of the line m must be equal to
m=4/3
I will assume a point (3,4)
Find the equation in slope intercept form
y=mx+b
we have
m=4/3
point (3,4)
substitute
4=(4/3)*(3)+b
solve for b
4=4+b
b=0
therefore
y=(4/3)x -----> equation in slope intercept form ( this line is perpendicular to line k)Find the equation in point slope form
y-y1=m(x-x1)
we have
m=4/3
point (3,4)
substitute
y-4=(4/3)*(x-3) -----> equation in point slope form ( this line is perpendicular to line k)Find the equation in standard form
Ax+By=C
where
A is a positive integer
B and C are integers
we have
y=(4/3)x
Multiply by 3 both sides
3y=4x
4x-3y=0 ------> equation in standard formy=(4/3)xy-4=(4/3)*(x-3)4x-3y=03x-22x + 1A8If x=8, then the length of AB is
Ok the length of AB is given by the equation:
[tex]3x-2[/tex]If x=8 then we simply have to replace this value in the equation:
[tex]\bar{AB}=3\cdot8-2=22[/tex]And that's the length of segment AB.
where does the x-intercept in to the y-intercept
x-intercept -3
y-intercept 6
15 lb of beans are distributed equally into 10 bags that give out of at the food bank how many pounds of beans are in each bag until your answer in simplest form
Determine the pounds of beans in each bag.
[tex]\begin{gathered} \frac{15}{10}=\frac{3\cdot5}{2\cdot5} \\ =\frac{3}{2} \\ =1\frac{1}{2} \end{gathered}[/tex]So answer is 1 1/2.
A ball is shot out of a cannon at ground level. it's height H in feet after t seconds is given by the function H(t) = 96t - 16t^2. Find H(1), H(5), H(2), and H(4). Why are some of the outputs equal? H(1) = ______ feetH(2)= ______ feetH(4)= ______ feet H(5)= ______ feet
Follow the function we have that
[tex]\begin{gathered} H(1)=96(1)-16(1)^2 \\ =96-16=80 \end{gathered}[/tex]So H(1) = 80 feet. Now
[tex]\begin{gathered} H(2)=96(2)-16(2)^2 \\ =192-64=128 \end{gathered}[/tex]So H(2) = 128 feet. Now
[tex]\begin{gathered} H(4)=96(4)-16(4)^2 \\ =384-256=128 \end{gathered}[/tex]So H(4) = 128 feet. Now
[tex]\begin{gathered} H(5)=96(5)-16(5)^2 \\ =480-400=80 \end{gathered}[/tex]So H(5) = 80 feet.
five to the third power
We need to find the value of 5 to the third power, to do this let's remember that the third power of a number means multiplying this number three times by itself, that is:
[tex]5^3=5\times5\times5=125[/tex]The answer is 125
Watch help videoFind the value of y in the diagram below.Y +9Y + 9Y +9y +9Y +9116Answer: Submit Answer
The equation from the box obtained is
[tex]y+9+y+9+y+9+y+9+y+9=116[/tex][tex]5y+45=116[/tex][tex]5y=71[/tex][tex]y=14.2[/tex]Simply this expression 4(1-3x)+7x-8
Answer:
-5x-4
Step-by-step explanation:
4(1-3x)+7x-8
Distribute the 4
4(1)+4(-3x)+7x-8
4-12x+7x-8
Combine like terms
-12x+7x+4-8
-5x-4
hopes this helps please mark brainliest
two trains leave town 906 miles aprt at the same time and travel each other. one train travels 21 mi/h faster than other. if they meet in 6 hours, what is the rate of each train?
Let
x be the speed of the slower train
x + 21 be the speed of the faster train
They meet in 6 hours which means that their speed is
[tex]\frac{906}{6}=151[/tex]The sum of their therefore is
[tex]\begin{gathered} x+(x+21)=151 \\ 2x+21=151 \\ 2x=151-21 \\ 2x=130 \\ \frac{2x}{2}=\frac{130}{2} \\ x=65\frac{\operatorname{mi}}{\operatorname{h}}\text{ (speed of slower train)} \\ \\ x+21=65+21=86\frac{\operatorname{mi}}{\operatorname{h}}\text{ (speed of faster train)} \end{gathered}[/tex]Find the principal P that corresponds to the future value F= $1,300 under r = 4% interest compounded daily for t = 3 years. Round your final answer to two decimal places.
The formula to find the principal is:
[tex]FV=P(1+i)^n[/tex]Where
FV is future value
P is principal
i is the rate of interest
n is the time frame
Given,
FV = 1300
i = r = 4%, or 4/100 = 0.04
n = t = 3
Plugging into formula and solving gives us:
[tex]\begin{gathered} FV=P(1+i)^n \\ 1300=P(1+0.04)^3 \\ 1300=P(1.04)^3 \\ P=\frac{1300}{1.04^3} \\ P=1155.695 \end{gathered}[/tex]Rounding to 2 decimal places:
$1155.70
Find the exact value of cos -1050.OA.-3OB.-12/2OC. 1OD. 1/3Reset Selection
Solution:
Given;
[tex]\cos(-1050)[/tex]Rewrite the expression using;
[tex]\cos(-x)=\cos x[/tex]Thus;
[tex]\begin{gathered} \cos(-1050)=\cos(1050) \\ \\ \cos(1050)=\cos(330) \end{gathered}[/tex]Then;
[tex]\cos(330)=\frac{\sqrt{3}}{2}[/tex]CORRECT OPTION: D
Jefferson works part time and earns 1,520in four weeks how much does he earn each weet
To determine how much he earns each week we need to divide the amount given by 4, that is we need to kae the division:
[tex]1520\div4[/tex]The long division is shown below:
To make the long division we notice that we can't divide the first number (one) by four, then we need to put down the five to get a 15, this number can be divided by four. Fifteen can be divided by four, the number four fits 3 times in fifteen, then we have a three as the first nonzero number. three by 4 is 12. We subtract 12 from 15 to get 3 and the we downed the following two. This procedure is repeated until we have a number that gives zero as a remainder. This is shown in the picture above.
From it we conclude that Jefferson earns $380 each week
The table lists recommended amounts of food to order for 25 party guests. Amanda and Syndey are hosting a graduation party for 40 guests. They know there will also be guests stopping by who may have come from other parties. For ordering purposes, they will count each of these "drop-in" guests as half a guest. How much of each food item should Amanda and Syndey order for a graduation party with 45 drop-in guests?
Amanda and Sydney are hosting a graduation party for 40 guests.
Also, they have 45 drop-in guests (each of these will count as half a guest).
Then, the total number of guests is:
[tex]40+\frac{45}{2}=\frac{40\times2+1\times45}{1\times2}=\frac{80+45}{2}=\frac{125}{2}[/tex]The table shows the recommended amounts of food for 25 party guests:
Fried Chicken: 24 pieces
Deli meats: 4 1/3 pounds
Lasagna: 10 3/4 pounds.
Let's find the proportion for 125/2 guests:
a. Fried chicken:
[tex]\begin{gathered} \frac{24\text{ pieces}}{25\text{ guests}}=\frac{x\text{ pieces}}{125/2\text{ guests}} \\ \text{Set the product of the diagonals equal to each other:} \\ 24\times\frac{125}{2}=x\times25 \\ \frac{24\times125}{2}=x\times25 \\ 1500=x\times25 \\ \text{Divide both sides by 25} \\ \frac{1500}{25}=\frac{x\times25}{25} \\ \text{Simplify} \\ 60=x \end{gathered}[/tex]Then, they'll need to order 60 units of fried chicken for the party.
b. Deli meats:
Start by converting the mixed number 4 1/3 into a fraction:
[tex]4\frac{1}{3}=\frac{4\times3+1}{3}=\frac{12+1}{3}=\frac{13}{3}[/tex]Now, apply proportions:
[tex]\begin{gathered} \frac{13/3\text{ pounds}}{25\text{ guests}}=\frac{x\text{ pounds}}{125/2\text{ guests}} \\ \text{Set the product of the diagonals equal to each other:} \\ x\times25=\frac{13}{3}\times\frac{125}{2} \\ x\times25=\frac{13\times125}{3\times2} \\ x\times25=\frac{1625}{6} \\ \text{Divide both sides by 25} \\ \frac{x\times25}{25}=\frac{1625}{6\times25}=\frac{1625}{150}=\frac{65}{6} \\ \text{Simplify} \\ x=\frac{65}{6} \end{gathered}[/tex]You also can convert this fraction into a mixed number:
[tex]\begin{gathered} \text{When you divide 65/6 you obtain a quotient of 10 and remainder 5.} \\ \text{Then the whole part is 10 and the fraction is 5/6} \\ \frac{65}{6}=10\frac{5}{6} \end{gathered}[/tex]Then, they'll need to order 10 5/6 pounds of deli meats for the party.
c. Lasagna
Convert the mixed number to fraction:
[tex]10\frac{3}{4}=\frac{10\times4+3}{4}=\frac{40+3}{4}=\frac{43}{4}[/tex]Apply proportions:
[tex]\begin{gathered} \frac{43/4\text{ pounds}}{25\text{ guests}}=\frac{x\text{ pounds}}{125/2\text{ guests}} \\ \text{Set the product of the diagonals equal to each other:} \\ x\times25=\frac{43}{4}\times\frac{125}{2} \\ x\times25=\frac{43\times125}{4\times2} \\ x\times25=\frac{5375}{8} \\ \text{Divide both sides by 25} \\ \frac{x\times25}{25}=\frac{5375}{8\times25}=\frac{5375}{200}=\frac{215}{8} \\ \text{Simplify} \\ x=\frac{215}{8} \end{gathered}[/tex]You also can convert this fraction into a mixed number:
[tex]\begin{gathered} \text{When you divide 215/8 you obtain a quotient of 26 and remainder 7.} \\ \text{Then the whole part is 26 and the fraction is 7/}8 \\ \frac{215}{8}=26\frac{7}{8} \end{gathered}[/tex]Then, they'll need to order 26 7/8 pounds of lasagna for the party.