Take x as the number of rows and y as the number of seats in each row.
There are 3300 seats in the auditorium, which means that the product of x and y is 3300:
[tex]x\cdot y=3300[/tex]We also know that y exceeds x by 16, this is:
[tex]y=x+16[/tex]Use these equations to find the number of seats and rows.
[tex]\begin{gathered} x\cdot(x+16)=3300 \\ x^2+16x=3300 \\ x^2+16x-3300=0 \end{gathered}[/tex]Use the quadratic equation to find x:
[tex]\begin{gathered} x=\frac{-16\pm\sqrt[]{16^2-4(1\cdot-3300)}}{2\cdot1} \\ x=\frac{-16\pm116}{2} \\ x1=\frac{-16+116}{2}=\frac{100}{2}=50 \\ x2=\frac{-16-116}{2}=\frac{-132}{2}=-66 \end{gathered}[/tex]As x is the number of rows, it can take negative values, so take the value of 50.
Use this values to find y:
[tex]\begin{gathered} x\cdot y=3300 \\ y=\frac{3300}{x} \\ y=\frac{3300}{50} \\ y=66 \end{gathered}[/tex]There are 50 seats in each row and there are 66 rows.
Two sides of a triangle are 5 and 15. Which of the following cannot be the third side? A. 6B. 12C. 19D. 20
Given:
Two sides of a triangle are 5 and 15.
To find:
Chose the correct option which cannot be the third side.
Explanation:
The length of a side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
Here, a sum of the two given sides is:
[tex]5+15=20[/tex]Also, the difference of two given sides,
[tex]15-5=10[/tex]So, the third side must be in between,
[tex]\begin{gathered} 15-5<\text{ third side < 15 +}5 \\ 10<\text{ third side < 20} \end{gathered}[/tex]Therefore, option (A) is not true according to the above property as 6 is less than 10.
Final answer:
Hence, (A) 6 cannot be the third side.
The radius of a circle is 8 kilometers. What is the area of a sector bounded by a 144° arc?Give the exact answer in simplest form. ____ square kilometers. (pi, fraction,)
Given the Circle below with radius r,
[tex]\text{Where r = 8km, }\phi=144^0\text{ and }\pi=\pi[/tex]The formula for the area, A, of the sector is given below as,
[tex]A=\frac{\phi}{360^0}\times\pi r^2[/tex]Substituting for r and the ∅ into the formula above,
[tex]\begin{gathered} A=\frac{144^0}{360^0}\times\pi\times8^2 \\ A=\frac{2}{5}\times\pi\times64=25\frac{3}{5}\pi km^2 \end{gathered}[/tex]Hence, Area A of the sector is 25 3/5 km²
What is the perimeter of thisquadrilateral?31.32 units41.2 units27.7 units50.8 units
We have to calculate the perimeter of the quadrilateral.
The perimeter is the sum of the lengths of the sides.
As we only know the coordinates of the vertices, we will have to calculate the length of each side as the distance between each endpoint.
We will label each vertex as:
(x1,y1) = (2, 6)
(x2,y2) = (4, 1)
(x3,y3) = (-4, -4)
(x4,y4) = (-3, 3)
We can then calculate each distance as:
[tex]\begin{gathered} d_{12}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d_{12}=\sqrt{(4-2)^2+(1-6)^2} \\ d_{12}=\sqrt{2^2+(-5)^2} \\ d_{12}=\sqrt{4+25} \\ d_{12}=\sqrt{29} \end{gathered}[/tex][tex]\begin{gathered} d_{23}=\sqrt{(x_3-x_2)^2+(y_3-y_2)^2} \\ d_{23}=\sqrt{(-4-4)^2+(-4-1)^2} \\ d_{23}=\sqrt{(-8)^2+(-5)^2} \\ d_{23}=\sqrt{64+25} \\ d_{23}=\sqrt{89} \end{gathered}[/tex][tex]\begin{gathered} d_{34}=\sqrt{(x_4-x_3)^2+(y_4-y_3)^2} \\ d_{34}=\sqrt{(-3-(-4))^2+(3-(-4))^2} \\ d_{34}=\sqrt{1^2+7^2} \\ d_{34}=\sqrt{1+49} \\ d_{34}=\sqrt{50} \end{gathered}[/tex][tex]\begin{gathered} d_{41}=\sqrt{(x_1-x_4)^2+(y_1-y_4)^2} \\ d_{41}=\sqrt{(2-(-3))^2+(6-3)^2} \\ d_{41}=\sqrt{5^2+3^2} \\ d_{41}=\sqrt{25+9} \\ d_{41}=\sqrt{34} \end{gathered}[/tex]Now we can add the four distances to find the perimeter:
[tex]\begin{gathered} P=d_{21}+d_{32}+d_{43}+d_{14} \\ P=\sqrt{29}+\sqrt{89}+\sqrt{50}+\sqrt{34} \\ P\approx5.39+9.43+7.07+5.83 \\ P\approx27.72 \end{gathered}[/tex]Answer: 27.7 units [Third option]
A quantity that varies or changes
Given: A quantity that varies or changes
Required: Name of the quantity.
Explanation:
A quantity that changes or varies as per the situation is called a "variable".
It is generally denoted by small alphabets, 'x', 'y', 'z' etc.
Final Answer: A quantity that varies or changes is called a variable.
Kofi and Ama together invest 52million in a business and agree to share the profit in the ratio of their investment. kofi received 5million and Ama received 8million as profit at the end of the first year. How much did each invest
let their ratio of investment
[tex]x\colon y[/tex]then
[tex]x+y=52[/tex]again
[tex]\begin{gathered} \frac{x}{y}=\frac{5}{8} \\ 8x=5y \end{gathered}[/tex]Now
[tex]\begin{gathered} 8x+8y=416 \\ 5y+8y=416 \\ 13y=416 \\ y=32 \end{gathered}[/tex]So
[tex]x=\frac{5\times32}{8}=20[/tex]So kofi invest 20 million and ama invest 32 million each
Question 1 Directions: Find the sector area for the circle below. Round to the 145° 18 ft 2
Do you have a picture of your question?
ok
angle = 145
radius = 18 ft
[tex]\begin{gathered} \text{Area = }\frac{\theta}{360}pi(r)^2 \\ \text{Area = }\frac{145}{360}(3.14)(18)^2 \\ \text{Area = (0.402)(3.14)(324)} \\ \text{Area = 409 ft}^2 \end{gathered}[/tex]Well the result from the calculator is 409.77 ft^2
I need to round to the near tenth or hundredth?
Rounding to the nearest tenth = 409.8 ft^2
Rounding to the nearest hundred = 409.77 ft^2
Find the area of the shaded region:102a3c5 cm2a7 cm
We want to find the shaded area. We could do this taking several ways. This time we are going to find the big area using two rectangles:
We will find the area for both and then add them:
area 1 + area 2
But inside of it there is an unshaded square.
That is why, we are going to substract the unshaded area:
This is
area 1 + area 2 - area 3
It will give us the shaded area.
Area 1We know that the area of a rectangle is given by
side 1 x side 2
In this case:
side 1 = 5cm
and
side 2 = 3 cm
Then, the area of the rectangle 1 is:
side 1 x side 2 = 5cm x 3 cm
= 15 cm²
area 1 = 15 cm²
Area 2We use
side 1 x side 2
For the second rectangle, we have:
Sue owes the bank more than $20. Use , or = to make the statement true. Sue's account value -$20
ANSWER
Sue's account value = -$20
EXPLANATION
If Sue owes the bank money, it means that her account value is less than 0. If she owes $20, then her account value is -$20.
5÷ 1/2=there are halves in one wholethere are halves in 5 wholes 5 is 1/2 of what number?
Answer:
There are 10 halves in 5 wholes
and
5 is 1/2 of 10
there are 2 halves in one whole
[tex]5\text{ }\div\text{ }\frac{1}{2}=10[/tex]Explanation:
Given the operation;
[tex]5\text{ }\div\text{ }\frac{1}{2}[/tex]To solve;
[tex]5\text{ }\div\frac{1}{2}=5\times\frac{2}{1}=\frac{5\times2}{1}=\frac{10}{1}=10[/tex]Therefore;
[tex]5\text{ }\div\text{ }\frac{1}{2}=10[/tex]There are 10 halves in 5 wholes
and
5 is 1/2 of 10
there are 2 halves in one whole
Suppose you wanted to invest $2500 with a small company owned by a friend of yours that pays 12.2% simple interest on the investment. How long would it take the money to grow to $3500 in this case? Show your work.
We have the following equation
[tex]3500=2500+2500\cdot0.122\cdot t[/tex]where the t is the time we want to know
[tex]\begin{gathered} 1000=2500\cdot0.122\cdot t \\ 1000=305t \\ t=\frac{1000}{305}=3.2786 \end{gathered}[/tex]so we have to wait at least 3 months to our money to increase to 3500
17. Josh has7yd of fabric. To complete his craft. Project he needs 3 times that amount how 8many yards of fabric does Josh need?
SOLUTION
Write out the information given
Josh has
[tex]\frac{7}{8}\text{yards of fabric}[/tex]Josh needs 3 times the fabric he has to complete his project
Hence
Josh will need
[tex]\begin{gathered} 3\times\frac{7}{8}\text{ yards of fabric for his project } \\ \\ \frac{3\times7}{8}=\frac{21}{8}=2\frac{5}{8}\text{yards of fabric} \end{gathered}[/tex]Hence Josh will require 2(5/8) yards of fabric to complete his craft
The first option (A ) is correct
A 1.6-m - tall women stands next to the Eiffel Tower. At this time of the day, her shadow is 0.5 m long. At the same time, the tower’s shadow is 93.75 m long. How tall is the Eiffel Tower?
We can graph the situation as follows:
Since the shadow must have the same angle, we can use the tangent of this angle to find the height of the tower.
Then, we have that:
[tex]\tan (\alpha)=\frac{1.6}{0.5}\Rightarrow\tan (\alpha)=3.2\Rightarrow arctan(\tan (\alpha))=\arctan (3.2)[/tex]Then
[tex]\alpha=\arctan (3.2)\Rightarrow\alpha=72.64[/tex]Thus, we can use this angle to find the height of the Eiffel Tower:
[tex]\tan (72.64)=\frac{h}{93.75}\Rightarrow h=93.75\cdot3.2\Rightarrow h=300[/tex]Then, the height of the Eiffel Tower, according to this, is 300 m.
6. Select all the equations that are equivalent to the equation 3x - 4 = 5.A. 3x = 9B. 3x - 4 + 4 = 5 + 4C. X-4 = 2D. x = 9E. -45-33
The given equation is
[tex]3x-4=5[/tex]Adding 4 on both sides, we get
[tex]3x-4+4=5+4[/tex]The given equation is equivalent to option B
Consider
[tex]3x-4+4=5+4[/tex]Solve
[tex]3x=9[/tex]The given equation is equivalent to option A
Consider
[tex]3x=9[/tex]
Dividing by 3 , we get
[tex]\frac{3x}{3}=\frac{9}{3}[/tex][tex]x=3[/tex]The given equation is not equivalent to option D.
Options C and E are also not equivalent to the given equation.
Hence the correct options are A and B.
. Frankie has $2.75 in coins, all in dimes and nickels. The total number of coinsis 35. If he has d dimes and n nickels, which system models the situation?A)d+n=3510d+5n=2.75B)d+n=2.7510d+5n=35C)d+n=2.755d+10n=35D)d+n=3510d+5n=275
d + n = 35
10d + 25n = 275
Explanation:Note that:
1 dime = $0.1
1 quarter = $0.25
Let the number of dimes be d
Let the number of quarters be n
Frankie has $2.75
0.1d + 0.25n = 2.75
Multiply through by 100
10d + 25n = 275...................(1)
Total number of coins is 35
d + n = 35
Therefore, the system of equations that models the situation is:
d + n = 35
10d + 25n = 275
Jun bought a t-shirt cost P150.00, The salesman gave a 20% discount to him which is equal to P30.00 Which of these is the rate?A. P150.00 C. 20%B. P30.00 D. 30%
We have to identify the rate.
As the discount is P$ 30 over an P$ 150, this correspond to a rate of:
[tex]\frac{30}{150}=0.20=20\%[/tex]Answer: 20% [Option C]
Carol Ryan is a pet groomer whoseyearly gross pay is $14,600. She does notclaim any allowances, and the state taxrate is 3.5%. How much does she pay instate taxes each month?
We have that the total gross pay is $14,600.
The state tax rate is 3.5%.
First, we need to find the 3.5% of $14,600:
[tex]14600\cdot\frac{3.5}{100}=\frac{51100}{100}=511[/tex]This is the total value of tax in a year. Then, to find how much she pays we need to divide this amount by 12 (number of months of the year). Then, we have:
[tex]T=\frac{511}{12}\Rightarrow T=42.583333[/tex]Therefore, she needs to pay $42.58 a month in state taxes.
In trapezoid ABCD, sides AB and DC are on parallel lines. A regular hexagon PQRSTU lies inside of trapezoid ABCD so that vertices P and Q trisect the base AB (with P closer to A and Q closer to B), S and T lie on the base CD, and sides PU and QR are parallel to sides AD and BC, respectively. What fraction of the area of trapezoid ABCD lies inside of hexagon PQRSTU? Express your answer as a common fraction.
To solve this problem, we have to divide the trapezoid in regions where we can estimate its area, proportional to another known area.
We can draw:
We can calculate the area of the hexagon as:
[tex]A_H=\frac{3\sqrt[]{3}}{2}s^2[/tex]Being s the side, for example the segment PQ. This measure (s) will be our relative measure to calculate all the other areas.
Then, we can calculate the area of the parallellograms APXD and QBCY. Both have the same area.
The base is equal to s, as the segments AP and QB are equal to PQ.
The height is equal to 2 times the apothem of the hexagon, that can be calculated as:
The apothem for a regular hexagon is:
[tex]a=\frac{\sqrt[]{3}}{2}s[/tex]So we can calculate the area of the parallelograms as:
[tex]A_P=b\cdot h=s\cdot(2a)=s\cdot2\frac{\sqrt[]{3}}{2}s=\sqrt[]{3}s^2[/tex]Now, we are only left with triangles UTX and SRY.
Its side is equal to the side of the hexagon and its height is equal to the apothem, so its area is equal to 1/6 of the hexagon.
We can calculate its area as:
[tex]A_T=\frac{b\cdot h}{2}=\frac{s\cdot\frac{\sqrt[]{3}}{2}s}{2}=\frac{\sqrt[]{3}}{4}s^2[/tex]Now, we can define the area of the trapezoid as the sum of the areas of the hexagon, 2 parallellograms and 2 triangles:
[tex]\begin{gathered} A_{Tz}=A_H+2A_P+2A_T \\ A_{Tz}=\frac{3\sqrt[]{3}}{2}s^2+2(\sqrt[]{3}s^2)+2\frac{\sqrt[]{3}}{4}s^2 \\ A_{Tz}=(\frac{3}{2}+2+\frac{2}{4})\cdot\sqrt[]{3}s^2 \\ A_{Tz}=(\frac{3+2\cdot2+1}{2})\cdot\sqrt[]{3}s^2 \\ A_{Tz}=(\frac{8}{2})\sqrt[]{3}s^2 \\ A_{Tz}=4\sqrt[]{3}s^2 \end{gathered}[/tex]We can now calculate the fraction of the area of trapezoid ABCD lies inside of hexagon PQRSTU dividing the area of the hexagon by the area of the trapezoid:
[tex]\frac{A_H}{A_{Tz}}=\frac{\frac{3}{2}\sqrt[]{3}s^2}{4\sqrt[]{3}s^2}=\frac{\frac{3}{2}}{4}=\frac{3}{2}\cdot\frac{1}{4}=\frac{3}{8}[/tex]Answer: The fraction of the area of trapezoid ABCD lies inside of hexagon PQRSTU is 3/8.
The Vols scored 14 points less than 3 times the Titans score this weekend. If the sum of their points was 70, find how many each team scored.
We can define some notation at first for this problem. Let V the score for The Vols and T for the Titans last weekend. From the information given we have the following conditions:
[tex]V+T=70\text{ (1)}[/tex][tex]V=3T-14\text{ (2)}[/tex]Now if we replace euqation (2) into (1) we have:
[tex]3T-14+T=70[/tex]And if we solve for T we got:
[tex]4T=84[/tex][tex]T=\frac{84}{4}=21[/tex][tex]V=(3\cdot21)-14=49[/tex]For each value of », determine whether it is a solution to 3 < 2r - 7.
we have the following inequality:
[tex]3<2r-7[/tex]let's solve for r as follows:
step 1. add 7 to both sides
[tex]\begin{gathered} 3+7<2r-7+7 \\ 10<2r \end{gathered}[/tex]step 2. divide both sides by 2
[tex]\begin{gathered} \frac{10}{2}<\frac{2r}{2} \\ 5From the above we have that r > 5 , thus, any value greater than 5 is a solution.Therefore:
8 is a solution
5 is not a solution
-5 is not a solution
-9 is not a solution
Factor out the coefficient of the variable term. The expression 3 z + 1 factored is
What Is the perimeter of a square with a side length of 8 inches
2p= 32"
1) The perimeter (2p) is the sum of the sides.
2) Since a square has four congruent sides, and each one, in this case, measures 8 "
Then we can write:
2p = 8 +8 + 8+8
2p = 32 inches
3) So the Perimeter is 32"
The number of houses in Central Village, New York, grows every year according to the function H(t) = 540(1.039)^t . where H represents the number of houses, and i represents the number of years since January 1995. A civil engineering firm has suggested that a new, larger well must be built by the village to supply its water when the number of houses exceeds 1.000. During which year will this first happen?
Factor 10s + 25.Write your answer as a product with a whole number greater than 1.
10s+25
Factor by 5:
5 (2s+5)
Is (7, 2) a solution to this system of inequalities? 4x + 7y > 13 2x + y 2 19
You can also check the solution by putting the value of x and y in the equation to get the equality.
A catapult hurls a cantaloupe from a heightof 12 feet at an initial velocity of 47 feetper second.n(t) = -16 t2 + 47t +12
To find the equation we use:
[tex]h(t)=-\frac{1}{2}gt^2+v_0t+h_0[/tex]Where h is the gravity, v0 is the intial velocity and h0 is the initial height.
Then, we replace the values,
[tex]\begin{gathered} g=32\frac{ft}{s^2};v_0=47\frac{ft}{s};h_0=12ft \\ h(t)=-\frac{1}{2}32t^2+47t+12 \\ h(t)=-16t^2+47t+12 \end{gathered}[/tex]What is the value of x that makes the following equation true? 3/4 x + 2 = -4
The value of x that satisfy the equation is x=-8
Which of the following scatterplots represents the data shown below?(1,26), (2, 33), (3, 15), (4, 4), (5, 30), (6, 35), (7, 9), (8, 35), (9, 11), (10, 19), (11, 17), (12, 13)
the scatter plot that represents the data is option B
Explanation:Given:
The ordered pairs of a scatter plot:
(1,26), (2, 33), (3, 15), (4, 4), (5, 30), (6, 35), (7, 9), (8, 35), (9, 11), (10, 19), (11, 17), (12, 13)
To find:
the graph that represents the data above
The x values of the datatset: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
The y values pof the dataset: {26, 33, 15, 4, 30, 35, 9, 35, 11, 19, 17, 13}
The highest y value is 35
Options A, options C and options D have y values that are above 40
Only option B has the maximum value of y below 40
Checking other points, we will see the scatter plot that represents the data is option B
I need help, can you help me, please and thank you!
To determine the lenght in the blueprint we divide the measurements in feet by 8 then:
Living room:
[tex]3\times4[/tex]Kitchen:
[tex]3\times3[/tex]Office:
[tex]3\times2[/tex]Bedroom:
[tex]3\times2[/tex]Bedroom:
[tex]4\times4[/tex]Bathroom:
[tex]4.5\times5[/tex][tex] {(v + 3)}^{2} + 4 =0[/tex]I don't understand
we will solve the following equation
[tex](v+3)^2+4=0[/tex][tex](v+3)=\pm\sqrt[]{-4}[/tex][tex](v+3)=\pm2i[/tex][tex]v=-3\pm2i[/tex]if the equation has the two numbers like that witch one would I use to solve the problem or would I do -10 + 30 and use that number to do 0+30 kinda like that
SOLUTION
When x = 0, y is
[tex]\begin{gathered} y=-10(0)+30 \\ y=0+30 \\ y=30 \end{gathered}[/tex]When x = 1, y is
[tex]\begin{gathered} y=-10(1)+30 \\ y=-10+30 \\ y=20 \end{gathered}[/tex]When x = 3, y is
[tex]\begin{gathered} y=-10(3)+30 \\ y=-30+30 \\ y=0 \end{gathered}[/tex]So the table is shown below