1/3 = 0.333333333333
Then
3.33333333 is a whole number
= 3 + 1/3
Translate to a system Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Given:
Let x be the number of the cards and y be the number of the package.
Given that the number of cards is at least 4 more than twice the number of packages.
[tex]x=2y+4[/tex]Given that mailing costs no more than $500 and the cost of mailing a card is $3 and for a package, the cost is $7.
[tex]3x+7y=500[/tex]Substitute x=2y+4 in this equation, we get
[tex]3(2y+4)+7y=500[/tex][tex]6y+8+7y=500[/tex][tex]13y=500-8[/tex][tex]y=\frac{492}{13}[/tex][tex]y=37.8[/tex]Let y=37 and substitute in x=2y+4, we get
[tex]x=2\times37+4[/tex][tex]x=78[/tex]Hence the number of cards = 78 and the number of packages =37.
The total cost for this is $493 not more than $500.
prove that 1+3+5+......2n-1=n²
As given by the question
There are given that the series
[tex]1+3+5+\cdots+(2n-1)=n^2[/tex]Now,
For step 1:
Put n=1
Then LHS =1
And
[tex]\begin{gathered} R\mathrm{}H\mathrm{}S=(n)^2 \\ =(1)^2 \\ =1 \end{gathered}[/tex]So,
[tex]\therefore L.H.S=R.H.S[/tex]P(n) is true for n=1.
Now,
Step 2:
Assume that P(n) istrue for n=k
Then,
[tex]1+3+5+\cdots+(2n-1)=k^2[/tex]Adding 2k+1 on both sides
So, we get:
[tex]1+3+5\ldots+(2k-1)+(2k+1)=k^2+(2k+1)=(k+1)^2[/tex]P(n) is true for n=k+1
By the principle of mathematical induction P(n) is true for all natural numbers n.
Hence,
[tex]1+3+5+\cdots+(2n-1)=n^2[/tex]For all n.
Hence proved.
h(x) =x² +9 if h(x)=9 , x =
The given expression as; h(x) =x² +9
for h(x) = 9
Substitute the value of h(x) = 9 in the given expression;
h(x) =x² +9
9 =x² +9
x² = 9 - 9
x² = 0
x = 0
Answer : x = 0
Using the hottest and coolest months data, find the equation for line of best fit for this data showing all steps by hand.
Let
x -----> average temperature
y ----> Electricity Bill
we take the points
(99,150) and (69,80)
step 1
Find out the slope
m=(80-150)/(69-99)
m=-70/-30
m=7/3
step 2
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=7/3
point (69,80)
substitute and solve for b
80=(7/3)(69)+b
b=80-161
b=-81
the equation is
y=(7/3)x-81
using a graphing tool
Remember that the value of y cannot be a negative number
Write each fraction in terms of the LCD.x2x + 12x - 1x + 13x22x – 111X + 1X + 13Need Help?Watch ItAdditional Materials
The given fractions are,
[tex]\frac{x^2}{2x-1},\text{ }\frac{x+1}{x+13}[/tex]The LCD of fractions is the least common multiple of the denominators.
So, the LCD of the above fractions is,
[tex](2x-1)(x+13)[/tex]Multiplying the numerator and the denominator of the fraction by a common term does not change the fraction.
So, the first fraction can be expressed in terms of the LCD as,
[tex]\frac{x^2}{2x-1}=\frac{x^2(x+13)}{(2x-1)(x+13)}[/tex]The second fraction can be expressed in terms of the LCD as,
[tex]\frac{x+1_{}^{}}{x+13}=\frac{(x+1)(2x-1)}{(2x-1)(x+13)}[/tex]Find the height of the cliff. If necessary, round to the nearest hundredth yard.
We are given a diagram showing a slope, and a vertical height. We now have what represents a right angled triangle. The distance from the base of the cliff to the end of the slope is given as 24 yards. The slope itself is 37 yards. We shall now determine the height of the cliff (from ground to top) as indicated.
Note that we shall use the Pythagoras' theorem which is;
[tex]c^2=a^2+b^2[/tex]Where we have
[tex]\begin{gathered} c=\text{hypotenuse (longest side)} \\ a,b=\text{other sides} \end{gathered}[/tex]We can now substitute the given values/side lengths and we'll have;
[tex]37^2=24^2+b^2[/tex][tex]1369=576+b^2[/tex]Subtract 576 from both sides;
[tex]793=b^2[/tex]Take the square root of both sides;
[tex]\begin{gathered} \sqrt[]{793}=\sqrt[]{b^2} \\ 28.160255\ldots=b \end{gathered}[/tex]Rounded to the nearest hundredth, the answer now becomes;
ANSWER:
[tex]b=28.16yd[/tex]The last option is the correct answer
(4.7 x 10-3) x 351Simplify the expressionusing scientific notation and express your answer(2.5 x 10') < (3.3 X 100)in scientific notation. Round your answer to the nearest thousandth.AnswerKeypadKeyboard Shortcutsx10
Given:
[tex]\frac{(4.7\times10^{-3})\times351}{(2.5\times10^5)\times(3.3\times10^6)}[/tex]Remove the brackets and multiply common terms
[tex]\begin{gathered} \frac{(4.7\times10^{-3})\times351}{(2.5\times10^5)\times(3.3\times10^6)} \\ =\frac{4.7\times10^{-3}\times351}{2.5\times10^5\times3.3\times10^6} \\ =\frac{4.7\times351\times10^{-3}}{2.5\times3.3\times10^5\times10^6} \\ =\frac{1649.7\times10^{-3}}{8.25\times10^{11}} \end{gathered}[/tex]Simplify further to get
[tex]\begin{gathered} \frac{1649.7\times10^{-3}}{8.25\times10^{11}} \\ =\frac{16497\times10^{-1}_{}\times10^{-3}}{825\times10^{-2}\times10^{11}} \\ =\frac{16497\times10^{-4}}{825\times10^9} \end{gathered}[/tex]This further gives
[tex]\begin{gathered} \frac{16497\times10^{-4}}{825\times10^9} \\ =\frac{16497}{825}\times\frac{10^{-4}}{10^9} \\ =19.996\times10^{-4-9} \\ =19.996\times10^{-14} \end{gathered}[/tex]Therefore, the answer is
[tex]19.996\times10^{-14}[/tex]Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 2 ≤ x ≤ 6.
To find the average rate of change over an interval we need to calculate how much the function has changed over that interval by subtracting the final value by the initial one and dividing by the lenght of the interval. With this in mind we have:
[tex]\begin{gathered} \text{rate}=\frac{19-13}{6-2} \\ \text{rate}=\frac{6}{4} \\ \text{rate}=1.5 \end{gathered}[/tex]The average rate of change for this interval is 1.5
A 51-inch TV suggests that the main diagonal of the TV is 51 inches. Determine the dimensions of the screen of a 51 -inch TV with a 16:9 aspect ratio.Please see attached photo
The aspect ratio 16:9 indicates the next relation between x and y:
[tex]\frac{y}{x}=\frac{16}{9}[/tex]Applying the Pythagorean theorem to the right triangle formed:
[tex]51^2=x^2+y^2[/tex]Isolating y from the first equation:
[tex]y=\frac{16}{9}x[/tex]Substituting in the second equation:
[tex]\begin{gathered} 51^2=x^2+(\frac{16}{9}x)^2 \\ 2601=x^2+(\frac{16}{9})^2x^2 \\ 2601=x^2+\frac{16^2}{9^2}^{}x^2 \\ 2601=x^2+\frac{256}{81}^{}x^2 \\ 2601=\frac{337}{81}^{}x^2 \\ 2601\cdot\frac{81}{337}=x^2 \\ 625.166172=x^2 \\ \sqrt[]{625.17}\approx x \\ 25\approx x \end{gathered}[/tex]Replacing in the equation of y:
[tex]\begin{gathered} y=\frac{16}{9}\cdot25 \\ y\approx44.44 \end{gathered}[/tex]The approximate dimensions are:
length = 25 in
height = 44.44 in
Simplify.4n + 12 + 7n4 n + 1923 n16 n +711 n+ 12
11 n+ 12
In this expression, to simplify means to reduce it to the simpler expression. Hence:
1) Grouping similar terms
4n + 12 + 7n =
2) Adding them up:
4n+7n+12=
11n +12
An online company is advertising a mixer on sale for 35% off the original price of $224.99 what is the sale price for the mixer? Round your answer to the nearest cent, if necessary.
Given:
The original price of mixer is $224.99.
The discount on the mixer is 35%.
Explanation:
Determine the discount amount on the mixer.
[tex]\begin{gathered} d=\frac{35}{100}\cdot224.99 \\ =78.7465 \end{gathered}[/tex]Determine the sale price of the mixer.
[tex]\begin{gathered} 224.99-78.7465=146.2435 \\ \approx146.24 \end{gathered}[/tex]So sale price of the mixer is $146.24.
Given the vectors u =-7j and w=-9i+4j, find 8u and u+w.Write your answers in the form ai+bj.
Recall that:
[tex]\begin{gathered} \text{For all a, b, c, d, e real numbers:} \\ (ai+bj)+(ci+dj)=(a+c)i+(b+d)j, \\ e(ai+bj)=(ea)i+(eb)j\text{.} \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} 8u=8(-7j)=8(0i-7j)=(8\cdot0)i+(8\cdot(-7))j=0i-56j=-56j\text{.} \\ u+w=(-7j)+(-9i+4j)=(0i-7j)+(-9i+4j)=(0-9)i+(-7+4)j \\ =-9i-3j\text{.} \end{gathered}[/tex]Answer:
[tex]\begin{gathered} 8u=-56j\text{.} \\ u+w=-9i-3j\text{.} \end{gathered}[/tex]Given quadrilateral MNPQ which of the following set of conditons would not be enough to know that MNPQ is a parrelogram?
For a shape to be considered a parallelogram it has to meet the following conditions:
0. The opposite sides must be equal
,1. The opposite sides are equal
,2. Adjacent sides are supplementary
,3. The diagonals bisect each other
,4. The opposite sides are parallel
For the quadrilateral to be considered a parallelogram then, the conditions that should be met are:
MN=QP and MQ=NP
MN || QP and MQ || NP
The diagonals MP and NQ bisect each other.
∠M=∠P and ∠N=∠Q
From the given options, the second one and the third one are not enough to determine MNPQ as a parallelogram
find the volume round to the nearest tenth use 3.14 for pi 5km
Step 1
List all parameters
[tex]\begin{gathered} \pi\text{ = 3.14} \\ r\text{ = 5km} \\ \end{gathered}[/tex]Step 2
Write the volume of a sphere
[tex]undefined[/tex]the line contains the point (-3,5) and is perpendicular to the line y=3x-4
two lines are perpendicular when the multiplication of their slopes is equal to -1. The slope of y = 3x - 4 is 3. Then the slope of a perpendicular line is:
[tex]\begin{gathered} m\cdot3=-1 \\ m=-\frac{1}{3} \end{gathered}[/tex]Slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. Replacing with point (-3, 5) and m = -1/3, we get:
5 = -1/3(-3) + b
5 = 1+ b
5 - 1 = b
4 = b
Then, the equation is:
y = -1/3x + 4
on a map, the scale is 5 cm = 2km what is the missing distance?town A distance to 5.6km is the actual distance
The distance on the map is 14 cm
Explanation:Parameters:
Map scale: 5 cm = 2 km
Given actual distance = 5.6km
Let x be the distance on map, then
x = 5.6 km
2x = 5 * 5.6
2x = 28
x = 28/2
= 14 cm
The 7th grade took a field trip to the zoo. 50 students rode in cars and the rest of the students were split equally onto 4 buses. There are 142 total 7th graders. How many students were on each bus?
traveledGiven:
The total number of students is N = 142.
The number of students riding in a car is n(C) = 50.
The total number of buses is b = 4.
The objective is to find the number of students traveling on each bus.
Explanation:
Consider the number of students travelled in each bus as s.
Then, the total number of students traveling in 4 buses will be 4s.
The algebraic expression for the total number of students N can be represented as,
[tex]N=n(C)+b(s)\text{ . . . . .(1)}[/tex]On plugging the given values in equation (1),
[tex]142=50+4s[/tex]On further solving the above equation,
[tex]\begin{gathered} 142-50=4s \\ 4s=92 \\ s=\frac{92}{4} \\ s=23 \end{gathered}[/tex]Hence, the number of students traveling on each bus is 23.
slove equations with variables on both sides-4k - 10 = -5k
We will investigate how to solve an equation consisting of one variable
We have the following equation at hand:
[tex]-4k\text{ -10 = -5k}[/tex]The basic rule applied in solving equation like above is mathematical operations. We apply basic operations like:
[tex]\text{adding, subtracting, multiplying, division}[/tex]on both sides of the equation accompained by a variable or a number in an attempt to isolate the variable ( k ).
To isolate the variable ( k ) we need all the terms involving the variable ( k ) on one side of the equation.
We will add ( 4k ) on both sides of the equation as follows:
[tex]\begin{gathered} -4k\text{ -10 + 4k= -5k + 4k} \\ (\text{ 4k - 4k ) - 10 = -k} \\ -10\text{ = -k} \end{gathered}[/tex]Now to remove the negative sign accompained by ( k ) on the right hand side of the equation. We wil multiply both sides with ( -1 ) as follows:
[tex]\begin{gathered} -1\cdot(-10)\text{ = -1}\cdot(-k) \\ 10\text{ = k} \end{gathered}[/tex]Hence, the value of ( k ) is:
[tex]10[/tex]
Write an expression for the operation described.
"5 divided by the product of 3 and 2"
A (5 ÷ 3) × 2(5 ÷ 3) × 2
B 3 × (2 ÷ 5)3 × (2 ÷ 5)
C (3 × 2) ÷ 5(3 × 2) ÷ 5
D 5 ÷ (3 × 2)
D] 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
The operation "5 divided by the product of 3 and 2" means that number 5 divided by the product of 3 and 2.
The mathematical representation of this operation is 5 ÷ (3 × 2).
The answer to this operation = 5 / 6.
Hence, 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
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Question 1 of 14, Step 1 of 10/19CorrectDetermine if the following expression is a polynomial.4 – 8x + x²AnswerKeyboaO Yes O No
Solution
Given
[tex]4-8x+x^2[/tex]We want to determine if it's a Polynomial
A polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Hence 4 - 8x + x^2 is a Polynomial
The table shows the weights of bananas at a grocery store. Complete the table so that there is a proportional relationship between the number of bananas and their weight.Number Of Bananas. Weight In Kilograms. 2 ? 0.72 15 ?
Let u make the first box x and the second box y.
If there is a proportional relationship between the number of bananas and their weights, it means that:
[tex]\frac{2}{x}=\frac{6}{0.72}=\frac{15}{y}[/tex]We can take the first pair and solve for x as follows:
[tex]\begin{gathered} \frac{2}{x}=\frac{6}{0.72} \\ 6x=2\times0.72=1.44 \\ x=\frac{1.44}{6} \\ x=0.24 \end{gathered}[/tex]We can solve for y in the same manner:
[tex]\begin{gathered} \frac{6}{0.72}=\frac{15}{y} \\ 6y=15\times0.72=10.8 \\ y=\frac{10.8}{6} \\ y=1.8 \end{gathered}[/tex]Therefore, the boxes are filled as shown below:
Answer this question
Okay, in this case the statement talks about the sum, according with this we need to find the sum of the number blue bikes (b) and 9 red bikes.
So, in this case the correct option is A. b+9 because it says sum
Given A = {(1, 3X-1, 5}(6, 4)), B = {(2, 0X4, EX-4, 5x0, 0)) and C = {(1, 1x0, 2x0, 3)(0, 4X-3, 5)), answer the following multiple
choice question:
From the list of sets A, B, and C above, choose the set of relations that correctly represents a function.
O Set A only
O Sets A and C only
O Sets A and B only
The functions is Set A and Set B.
What is meant by function?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
A relation in a set is said to be a function, if every first element of an ordered pair in a set is related with a unique element of a second element.
No, two distinct second elements of an ordered pair, has the same first element.
For, example, {(1,2), (1,3), (4,5)}, is not a function, but it is a relation.
In Ordered pair, (x, y)
x=First Element
y= Second Element
→In Set A
First Element Second Element
1 3
-1 5
6 4
Every First element of set A has a unique second element. So, it is a function.
→In Set B
First Element Second Element
2 0
4 6
-4 5
0 0
Every First element of set B has unique second element and no two distinct Second element of set B, has same first element. So, it is a function.
→In Set C
First Element Second Element
1 1
0 2
0 3
-3 5
As, two same first elements of set C has distinct second element. So, it is not a function.
Therefore, Set A and Set B, are functions .
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3. A toy box is 24 cm long, 15 cm wide and 11 cm high. What is the volume of the toy box? What is the correct number sentence for this problem? A.V=24×15×11B.V=24×15C.V=24×11D.V=15×11
ANSWER
[tex]\begin{gathered} V=24*15*11 \\ V=3960\text{ }cm^3 \end{gathered}[/tex]EXPLANATION
The box is a rectangular prism. The volume of a rectangular prism is given by:
[tex]V=L*W*H[/tex]where L = length
W = width
H = height
Therefore, the volume of the box can be written in the number sentence:
[tex]V=24*15*11[/tex]and the volume of the box is:
[tex]V=3960\text{ }cm^3[/tex]That is the answer.
the relationship between the minutes a candle is burned and the size of the candle in millimeters is shown on the graph.
The function is a decreasin line so the more time goes the side will decrease so the correct answer is:
The candle started at 9mm and shrinks 5mm every 4 minutes
What is the solution to the equation below? Round your answer to two decimal places.ln x = 0.2A.x = 1.58B.x = -0.70C.x = -1.61D.x = 1.22
Given the equation:
[tex]\ln \left(x\right)=0.2[/tex]Apply the properties of logarithms:
[tex]e^{ln(x)}=e^{0.2}[/tex]Simplify:
[tex]x=e^{0.2}=1.22[/tex]Answer: D. x = 1.22
Find the cardinal number of the setWhere N denotes the set of all natural numbers
And x is divisible by 6 . The natural number of x will be
[tex]x=\mleft\lbrace36,42,48,54\mright\rbrace[/tex]Annie's backyard deck cost $61.75 per square meter to build. The deck is 7 meters wide and14 meters long. How much did it cost to build the deck?
ANSWER
the cost to build the deck is $6051.5
EXPLANATION
Given that;
The length of the deck is 14 m
The width of the deck is 7m
1 m^2 is equivalent to $61.75
Follow the steps below to find the cost to build the deck
Step 1; Find the area of the deck
[tex]\begin{gathered} \text{ Recall, that the deck is a rectangular shape} \\ \text{ Area of a rectangle = length }\times\text{ width} \\ \text{ Area of a reactangle = 14 }\times\text{ 7} \\ \text{ Area of a rectangle = 98m}^2 \end{gathered}[/tex]Step 2; Find the total cost of the deck
Let x represents the total cost to build the deck
[tex]\begin{gathered} \text{ 1m}^2\text{ }\rightarrow\text{ \$61.75} \\ \text{ 98m}^2\text{ }\rightarrow\text{ \$x} \\ \text{ cross multiply} \\ \text{ 1m}^2\text{ }\times\text{ \$x = \$61.75 }\times\text{ 98m}^2 \\ \text{ Isolate \$x }\frac{}{} \\ \text{ \$x = }\frac{\text{ \$61.75}\times98\cancel{m^2}}{1\cancel{m^2}} \\ \text{ \$x = \$61.75 }\times\text{ 98} \\ \text{ \$x = \$6051.5} \end{gathered}[/tex]Therefore, the cost to build the deck is $6051.5
what is 12/8 × 18/16
First of all, simplify the given fractions
The current student population of Kansas City is 2700. If the population increases at a rate of 5.2% each year. What will the student population be in 4 years?Write an exponential growth model for the future population P(x) where x is in years:p(x)=What will the population be in 4 years? (Round to nearest student)
ANSWER
P(x) = 2700(1.052)^t
P(4) = 3307. (Rounded to nearest student)
EXPLANATION
Given:
1. The current student population to be 2700
2. The growth rate = 5.2% = 0.052
Desired Outcome
1. The exponential growth model
2. Population of the students in 4 years
The Exponential Growth Model
[tex]\begin{gathered} P(x)\text{ = 2700\lparen1 + 0.052\rparen}^t \\ P(x)\text{ = 2700\lparen1.052\rparen}^t \end{gathered}[/tex]Population in 4 years
[tex]\begin{gathered} P(4)\text{ = 2700\lparen1.052\rparen}^4 \\ P(4)\text{ = 2700}\times1.2248 \\ P(4)\text{ = 3306.96} \end{gathered}[/tex]Hence, the Exponential Growth Model P(x) = 2700(1.052)^t and the Population of the students in 4 years P(4) = 3307. (Rounded to nearest student)