In order to find the unit cost of each option, let's divide the cost by the can size.
So we have:
[tex]\begin{gathered} a\text{.} \\ \text{unit cost}=\frac{3.99}{12}=0.3325 \\ \\ b\text{.} \\ \text{unit cost}=\frac{6.29}{4\cdot6}=0.2621 \\ \\ c\text{.} \\ \text{unit cost}=\frac{3.39}{3\cdot3}=0.3767 \end{gathered}[/tex]The lower unit cost is from option b, so this is the best buy.
A lumber yard sells square scraps of plywood with sides varying from 1 foot to 4 feet. Ed wants to use some of thesepieces to build storage cubes. The relationship between the length of the side of a cube and the volume of the cubeis expressed by the functionf(x) = x³where x is the length of a side of the cube. What is the range of this function in cubic feet for the domain given?
Solution:
Given that the relationshoip between the length of the cube and its volume is expressed as
[tex]\begin{gathered} f(x)=x^3 \\ where \\ x\Rightarrow length\text{ of the cube} \\ \end{gathered}[/tex]The range of the above function is the dependent values for which the function is real.
Given that the domain of the function is from 1 foot to 4 feet, this implies that
[tex]\begin{gathered} f(x)=x^3 \\ when\text{ x=x1,} \\ f(1)=1^3 \\ \Rightarrow f(1)=1 \\ when\text{ x = 4} \\ f(4)=4^3 \\ \Rightarrow f(4)=64 \\ \\ \\ \\ \end{gathered}[/tex]Hence, the range of the function in cubic feet varies from 4 to 64
a submarine started at the surface of the ocean and descended 1.740 ft in 30 minutes what integer represents the change in depth depth and feet per minute
The submarine descends to 1.740ft
Time taken = 30mins
Change in depth = (1.740-0)/(30-0) =1.740/30 = 0.058 ft/min
An interger is a whole number that can be positive, negative or zero
As an interger, the change in depth is a
A cone with a fixed height of 15 inches is shown on a computer screen. An animator increases the radius r at a rate of 6 inches per minute. Which of the following gives the volume v(r) of the cone as a function of time f(t)? Assume the radius is 6 inches at t = 1.
Answer::
[tex]V(r)=60\pi rt[/tex]Explanation:
For a cone of radius r and height, h
[tex]\text{Volume}=\frac{1}{3}\pi r^2h[/tex]The cone has a fixed height of 15 inches.
The radius increases at a rate of 6 inches per minute.
We have that:
[tex]\begin{gathered} V=\frac{1}{3}\pi\times15r^2 \\ V=5\pi r^2 \end{gathered}[/tex]Taking the derivative with respect to time(t), we have:
[tex]\begin{gathered} \frac{dV}{dt}=5\pi2r\frac{dr}{dt} \\ Since\text{ }\frac{dr}{\mathrm{d}t}=6\text{ inches per minute} \\ \frac{dV}{dt}=5\pi2r\times6 \\ \frac{dV}{dt}=60\pi r \end{gathered}[/tex]We then rewrite in order to integrate.
[tex]\begin{gathered} dV=60\pi\text{rdt} \\ \int dV=\int 60\pi\text{rdt}=60\pi\int \text{rdt} \\ V=60\pi rt+C,C\text{ a constant of integration} \end{gathered}[/tex]Therefore, we have:
[tex]V(r)=60\pi rt[/tex]Martha borrowed $300 from a lender that charged simple interest at an annual rate of 7 % .When Martina paid off the loan , she paid $105 in interest.How long was the loan for ,in years?
ANSWER
5 years
EXPLANATION
The simple interest formula is,
[tex]i=P\times r\times t[/tex]Where
• i: interested accumulated
,• P: principal amount
,• r: annual interest rate
,• t: time in years
In this problem we know that the principal amount P = 300, the interest rate is r = 7% and the accumulated interest i = 105. We have to find the time t in years,
[tex]105=300\cdot0.07\cdot t[/tex]Solving for t,
[tex]t=\frac{105}{300\cdot0.07}=\frac{105}{21}=5[/tex]The loan was for 5 years.
Find the slope of the line containing the two points. (If an answer is undefined, enter UNDEFINED.)(0, 5), (4, 0)
The Slope of a Line
If we are given the coordinates of two points contained in a line, it is possible to calculate the line's slope by using the formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Where the coordinates of the two points are (x1,y1) and (x2,y2) respectively.
The points are (0, 5) and (4, 0). Substituting the corresponding values in the formula, we have:
[tex]m=\frac{0-5}{4-0}[/tex]Calculating:
[tex]\begin{gathered} m=\frac{-5}{4} \\ m=-\frac{5}{4} \end{gathered}[/tex]The slope is -5/4
Ellis opened a checking and a savings account on the same day. His checking account had an initial balance of $1,113.50, and his savings account had an initial balance of $200.25. Each week, he withdrew $50 from his checking account and deposited $20.25 into his savings account. If he made no other withdrawals or deposits, and the balance of the two accounts are currently equal, how many weeks ago did Ellis open the accounts?
For the checking account:
Let set x for each week when Ellis withdrew.
The initial balance is $1,113.50
and he withdrew $50 from this account.
Then, we can write the next equation:
C=1,113.50-50x
For the saving account:
Let set x for each week when Ellis deposited.
The initial balance is $200.25
and he deposited $20.25 each week.
Then, we can write the next equation:
S=200.25+20.25x
Currently, both accounts are equal. Therefore, we can equal both equations and find the x value to know where both accounts were opened.
Then:
C = S
1,113.50-50x = 200.25+20.25x
Solve for x:
1,113.50-200.25=25x+50x
913.25= 75x
x = 12.17
12.17=x
if the width of a pool is 60 ft and the perimeter is 200 ft. how long is the pool?A.40 ftB.50 ftC.140 ft.D.120 ft.
SOLUTION
If the width of a pool is 60 ft and the perimeter is 200 ft.
How long is the pool?
Given the perimeter = 200 ft
Width = 60 ft
Length = x
Perimeter = 2 ( L + W )
200 = 2 ( L + 60 )
Divide both sides by 2, we have :
100 = L + 60
L = 100 - 60
L = 40 ft.......................OPTION A
I need to analyze this graph and table and respond to this questionExplain what the origin represents in this situation.
1) Gathering the data
Pounds (Potatoes) | Price ($)
3 2.97
4 3.96
6 5.94
10 9.90
2) Let's try to find a slope for that, picking any pair of points (4,4) and (3,3)
m = y_2 -y_1 / x_2 - x_1
m=3-4/3-4
m=-1/-1
m=1
Plugging into the linear equation y=mx +b one of those points, to find the linear parameter (b) we have:
y=mx+b
3=1(3)+b
3=3+b Flipping it
b=0
So the function that best describes this flow of prices is y= x
3) The origin represents, looking back at the table. In the origin there's not potato to sell, so no one is paying for nothing. And the seller only gets paid by each pound he gets.
(pound) potatoes | $ price
0 0
3 2. 97 (approximately 3)
4 3.96 (approximately 4)
The sum of the measures of any two supplementary angles is 180 degree. < L and < S are supplementary angles. The measures of < L is sixty-six degrees less than five times the measure of < S.What is the measure of < L? Use “deg” for the degree symbol.What is the measure of < S? Use “deg” for the degree symbol.
If the sum of the measures of any two supplementary angles is 180 degrees and m[tex]m\angle L+m\angle S=180^0\text{ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 1}[/tex]Also, if the measure of < L is sixty-six degrees less than five times the measure of < S, then:
[tex]m\angle L=5m\angle S-66_{--------------}2[/tex]Substitute equation 2 into equation 1 to have:
[tex](5m\angle S-66)+m\angle S=180[/tex]Simplify the result to have:
[tex]\begin{gathered} 5m\angle S-66+m\angle S=180 \\ 5m\angle S+m\angle S-66=180 \\ 6m\angle S-66=180 \end{gathered}[/tex]Add 66 to both sides of the resulting equation:
[tex]\begin{gathered} 6m\angle S-66+66=180+66 \\ 6m\angle S=246 \end{gathered}[/tex]Divide both sides of the equation by 6
[tex]\begin{gathered} \frac{\cancel{6}m\angle S}{\cancel{6}}=\frac{\cancel{246}^{41}}{\cancel{6}} \\ m\angle S=41^0 \end{gathered}[/tex]Get the measure of < L. Recall that:
[tex]\begin{gathered} m\angle L=5m\angle S-66 \\ m\angle L=5(41)-66 \\ m\angle L=205-66 \\ m\angle L=139^0 \end{gathered}[/tex]Hence the measure of < L is 139 deg. while the measure of
How do you find linear eqations in standard form?
A linear equation in standard form is given by
[tex]Ax+By=C[/tex]Where x and y are variables and A, B and C are constants.
For Example:
[tex]3x+5y=2[/tex]6.) $75 is deposited monthly into an account that pays 1.8% interestcompounded monthly. What is the amount in the account after 3 years?
Answer
$142.55
Explanation
From the question, the following were given:
Principal, P = $75
Rate, R = 1.8% = 1.8/100 = 0.018/12 = 0.0015
Time, nt = 3 years x 12 months = 36 months
The amount, A in the account after 3 years can be calculated as follow
[tex]\begin{gathered} \text{A }=P(1+R)^{nt} \\ A=75(1+0.018)^{36} \\ A=75(1.018)^{36} \\ A=75\times1.900728156 \\ A=142.55 \end{gathered}[/tex]Therefore, the amount in the account after 3 years is $142.55
16A particle moves along the x-axis so that its velocity y at any given time t, for O≤† ≤16, is given byM(t) = e-*' -1. At time t=0, the particle is at the origin. During what intervals of time is the particlemoving to the left?Round to the nearest thousandth
We have an expression for the velocity of the particle:
[tex]v(t)=e^{2\sin t}-1[/tex]We have to find when the particle moves to the left.
As the particle moves along the x-axis, this means that the velocity is negative.
We then have to find the interval where v(t) < 0.
We can find this interval as:
[tex]\begin{gathered} v(t)<0 \\ e^{2\sin t}-1<0 \\ e^{2\sin t}<1 \\ \ln(e^{2\sin t})<\ln(1) \\ 2\sin t<0 \\ \sin t<0 \end{gathered}[/tex]The function sin(t) is negative for intervals between π and 2π per cycle.
As t is defined from 0 to 16, we can calculate how many cycles we have:
[tex]f=\frac{16}{2\pi}\approx2.546[/tex]We will have at least 2 intervals or 3 at most where sin(t) < 0.
We can list the intervals as:
[tex]\begin{gathered} (\pi,2\pi) \\ (3\pi,4\pi) \\ (5\pi,16) \end{gathered}[/tex]The third period is cut at t = 16.
We can skecth the velocity as:
We can round the intervals to the nearest thousand as:
[tex]\begin{gathered} (\pi,2\pi)=(3.142,6.286) \\ (3\pi,4\pi)=(9.425,12.566) \\ (5\pi,16)=(15.708,16) \end{gathered}[/tex]Answer: the intervals for t are (3.142, 6.286), (9.425, 12.566) and (15.708, 16).
Suppose you have a triangle, and you want to construct a similar triangle. Which twotransformations will result in a triangle that is similar, but is not congruent?Rotation and ReflectionReflection and TranslationReflection and DilationTranslation and Rotation
The key to understanding what the question requires is to pay atention to the details given to us in the question:
I. We have a triangle
II. We want to construct a similar triangle
III. The triangle in II is to be similar but not congruent (of identical sizes)
To achieve the following, we must understand the varying types of transformations available
Rotation means that the shape is turned about a fixed point (the center of rotation); the shape remains the same size, only its direction is changed
Reflection means that all the vertices of a figure are flipped (kind of like a mirror image); the shape remains the same size, only its direction is changed
Translation means that the figure is moved in a straight line; the shape remains the same size, its direction/orientation remains the same
Dilation means that the figure is of a different size; the shape remains the same, the size is different
Hence, the correct answer is Reflection and Dilation (option C)
49. Which statement is true for every solution of the system y>x+7 y+x>7?A. x > 6B. y > 6C. y > 7D. x ≤ 4
Given,
The equation of inequality is,
[tex]\begin{gathered} y>x+7 \\ x+y>7 \end{gathered}[/tex]Required
The solution of the equation.
Here, the value of y is greater than sum of x+7.
If x = 0 then the value of y is,
[tex]y>0[/tex]Similarly for expression x+y >7
If x = 0 then the value of y is,
[tex]y>7[/tex]Hence, the true statement for the given inequalities is y > 7.
Write both solutions3x² + 2x - 8 = 0
Given:
[tex]3x^2+2x-8=0[/tex]Required:
To find the solution of the given equation.
Explanation:
Consider
[tex]3x^2+2x-8=0[/tex][tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex][tex]\begin{gathered} =\frac{-2\pm\sqrt{2^2-4(3)(-8)}}{2(3)} \\ \\ =\frac{-2\pm\sqrt{4+96}}{6} \\ \\ =\frac{-2\pm10}{6} \end{gathered}[/tex]Now
[tex]\begin{gathered} x=\frac{-2+10}{6} \\ \\ x=\frac{8}{6} \\ \\ x=\frac{4}{3} \end{gathered}[/tex]And
[tex]\begin{gathered} x=\frac{-2-10}{6} \\ \\ x=\frac{-12}{6} \\ \\ x=-2 \end{gathered}[/tex]Final Answer:
[tex]x=-2,\frac{4}{3}[/tex]Complete the rate table. Enter fractional answers as decimals.Distance (km)0.4Time (min)510202560
Answer:
Distance (km) 0.4 0.8 1.6 2 4.8
Time (min) 5 10 20 25 60
Explanation:
This is a direct relation, it means that when the time increase, the distance increase. So, the ratio of the distance to the time is constant, then we can calculate the distance for 10 min as follows
[tex]10\text{ min}\times\frac{0.4\text{ km}}{5\text{ min}}=\frac{10\text{ min}\times0.4\text{ km}}{5\text{ min}}=0.8\text{ km}[/tex]In the same way, we get the following for 20, 25, and 60 min
[tex]20\text{ min}\times\frac{0.4\text{ km}}{5\text{ min}}=\frac{20\text{ min }\times0.4\text{ km}}{5\text{ min}}=1.6\text{ km}[/tex][tex]25\text{ min }\times\frac{0.4\text{ km}}{5\text{ min}}=\frac{25\text{ min }\times0.4\text{ km}}{5\text{ min}}=2\text{ km}[/tex][tex]60\text{ min}\times\frac{0.4\text{ km}}{5\text{ min}}=\frac{60\text{ min }\times0.4\text{ km}}{5\text{ min}}=4.8\text{ km}[/tex]Therefore, the complete table is
Distance (km) 0.4 0.8 1.6 2 4.8
Time (min) 5 10 20 25 60
Not sure if I got the right answer and I need help showing work please :(
The expression:
[tex](x-5)(x+5)[/tex]Looks like the factorization of a difference of squares, being those squares x^2 and 25. Then, correct answer wil be C, the only available option where we have a difference of squares. We can check this by performing the product:
[tex](x-5)(x+5)=x\cdot x+5x-5x-5\cdot5[/tex]We just performed "FOIL method" for the product of binomials. Multiplication of the First terms, then the Outer, then the Inner and finally the Last terms.
Reorganizing the expression, and simplifying like terms:
[tex]\begin{gathered} (x-5)(x+5)=x^2+5x-5x-25 \\ (x-5)(x+5)=x^2-25 \end{gathered}[/tex]Leon designs a fabric pattern using figure STUVW. He dilates figure STUVW using a scale factor of 2 with a center of dilation at vertex V to form figure STUVW. Then Leon rotates this image 90' clockwise around the origin to form figure STUVW". What are the coordinates of figure STUVW?
The formula for dilation about a point (h, k) is given by
[tex](x,y)\rightarrow(h+t(x-h),k+t(y-k))[/tex]We are dilating the figure by a factor of 2 about vertex V(0,1 ); therefore, the coordinate transfromations will be
[tex]S(0,5)\rightarrow S^{\prime}(0,9)_{}[/tex][tex]\begin{gathered} T(3,5)\rightarrow T^{\prime}(6,9) \\ U(3,1)\rightarrow U^{\prime}(6,1) \\ V(0,1)\rightarrow V^{\prime}(0,1) \\ W(2,3)\rightarrow W^{\prime}(4,5) \end{gathered}[/tex]Finally, roatiing these coordinates 90 degree clockwise gives us
[tex](x,y)\rightarrow(y,-x)[/tex][tex]\begin{gathered} S^{\prime}(0,9)\rightarrow\textcolor{#FF7968}{S^{\doubleprime}(9,0)} \\ T^{\prime}(6,9)\rightarrow\textcolor{#FF7968}{T^{\doubleprime}(9,-6)} \\ U^{\prime}(6,1)\rightarrow\textcolor{#FF7968}{U^{\doubleprime}(1,-6)} \\ V^{\prime}(0,1)\rightarrow\textcolor{#FF7968}{V^{\doubleprime}(1,0)} \\ W^{\prime}(2,3)\rightarrow\textcolor{#FF7968}{W^{\doubleprime}(3,-2)} \\ \end{gathered}[/tex]what is the value of 5 - 17?
When you did the operation you get
[tex]5-17=-12[/tex]The length, 1, of a wheelchair ramp is 3 more than 11 times itswidth, w. Which equation represents the relationship betweenthe length and the width of the ramp?
Let the length of the ramp is L, and the width is W
Since the length is 3 more than 11 times the width, then
[tex]\begin{gathered} L=11\times W+3 \\ L=11W+3 \\ \end{gathered}[/tex]Which of the following are solutions to the equation? 3x-4y-8 =12 Select all that apply.
The (0, -5), (4, -2), (-16, -17) are the solutions for the equation 3x-4y-8 = 12
The equation is 3x-4y-8 = 12
Consider the LHS and compare it with RHS
First point (0,-5)
3(0) -4(-5)-8 = 0+20-8
= 12
LHS = RHS
(4,-2)
3(4) - 4(-2)-8 = 12+8-8
= 12
LHS = RHS
(8,2)
3(8) - 4(2) -8 = 24-8-8
= 8
LHS ≠ RHS
(-16,-17)
3(-16) - 4(-17) -8 = -48+68-8
= 12
LHS = RHS
(-1,-8)
3(-1) - 4(-8) -8 = -3+32-8
= 21
LHS ≠ RHS
(-40,-34)
3(-40) - 4(-34) - 8 = -120+136-8
= 8
LHS ≠ RHS
Hence, the (0, -5), (4, -2), (-16, -17) are the solutions for the equation 3x-4y-8 = 12
The complete question is:
Which of the following are solutions to the equation? 3x-4y-8 =12 Select all that apply
(0, -5), (4, -2), (8, 2), (-16, -17), (-1, -8), (-40, -34)
Learn more about equation here
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Emily's meal costs $32.75 and Darren's ameal costs $39.88. Emily treats Darren bypaying for both meals, and leaves a 14%tip. Find the total cost.
Their meals were $32.75 and $39.88. So the total cost was:
32.75+39.88 = 72.63. She payed for both and left a 14%, 0.14, tip.
Let's calculate the tip:
0.14*72.63 = 10.1682.
So, the total cost will be 72.63+10.1682 = $82.80.
Evaluate the function: w(x) = -2x+1Find: w(-1)
When we evaluate a function w(x) we replace x by the given value.
so:
[tex]w(-1)=-2(-1)+1[/tex][tex]w(-1)=2+1\text{ = 3}[/tex]Find f of g.f(x) = x2 - 1, g(x) = x2 + 3
f of g = x⁴ + 6x² + 8
Explanation:
f(x) = x² - 1, g(x) = x² + 3
f o g(x) means we replace the x in f(x) with the g(x) function
f o g(x) = (x² + 3)² - 1
expanding the bracket:
f o g(x) = (x² + 3)(x² + 3) - 1
= x²(x² + 3) + 3(x² + 3) - 1
= x⁴ + 3x² + 3x² + 9 - 1
collect like terms:
f o g(x) = x⁴ + 6x² + 8
Hence, f of g = x⁴ + 6x² + 8
abus ? m Find the radius of a circle with a circumference of 57 m. Round your answer to the nearest tenth. ? v
The formula for the circumference is
[tex]L=2\cdot\pi\cdot r[/tex][tex]r=\frac{L}{2\pi}[/tex][tex]r=\frac{57}{2\cdot3.14}=9.07[/tex]The radius would be 9.07m
He claims that the two farms are similar because farm A can be mapped onto farm B through the translation (x x+6,; y y+4) and a dilation about the center by a scale factor of k. What is the value of k ?
Answer:
k = 4/7
Explanation:
The radius of farm A is 7 units and the radius of farm B is 4 units. Then, the scale factor can be calculated as the ratio between the radius, so the scale factor from farm A to farm B is 4/7
Therefore, k = 4/7
convert fraction to percent 3/15
To convert the fraction to percent multiply it by 100%
To change the fraction 3/15 to percent multiply it by 100%
[tex]\frac{3}{15}\times100=\frac{3\times100}{15}=\frac{300}{15}[/tex]Now divide 300 by 15
[tex]\frac{300}{15}=20[/tex]Then 3/15 = 20%
If the radius of a sphere is doubled, its surface area is multiplied byA. 2B. 4C. 27D. 16
Answer:
B. 4
Explanation:
The surface area of a sphere is calculated using the formula:
[tex]S_o=4\pi r^2\text{ where r=radius}[/tex]If the radius is doubled:
• The new radius, R=2r
We then have the new surface area as:
[tex]\begin{gathered} S_1=4\pi R^2=4\pi(2r)^2 \\ =4\pi(4r^2) \\ =4\times4\pi r^2 \\ S_1=4\times S_0 \end{gathered}[/tex]We see from the above (4 x So) that if the radius of a sphere is doubled, its surface area is multiplied by 4.
The correct choice is B.
A certain substance has a half-life of 2 minutes. A fresh sample of the substance weighing 60 mg was obtainedAfter how many minutes will there be 14 mg of the substance remaining?
Answer:
[tex]4.2\text{ minutes}[/tex]Explanation:
Here, we want to get the number of minutes it will take for 14 mg of the substance to remain
We can have an exponential equation that describes the scenario as follows:
where A(t) is the amount remaining after t minutes
and t is the number of minutes
[tex]\begin{gathered} A(t)=60(0.5)^{\frac{t}{2}} \\ (\frac{14}{60})^2=0.5^t \\ 0.054=0.5^t \\ \ln \text{ 0.054 = tln0.5} \\ t\text{ = }\frac{\ln \text{ 0.054}}{\ln \text{ 0.5}} \\ t\text{ = }4.2\text{ minutes} \end{gathered}[/tex]Solve the following system of equations-5x+2y=410x-2y=-14x=y=
Given
system of equations
-5x+2y=4
10x-2y=-14
Find
The values of x and y
Explanation
by elimination method we find the values of x and y
- 5x + 2y = 4
2(5x - y)= - 14
this implies = 5x - y = - 7
two equations are
- 5x + 2y = 4 .............(1)
5x - y = - 7 ............(2)
add both equations (1) and (2)
[tex]\begin{gathered} -5x+2y+5x-y=4+(-7) \\ y=-3 \end{gathered}[/tex]substitute the value of y in equation (1)
[tex]\begin{gathered} -5x+2(-3)=4 \\ -5x-6=4 \\ -5x=10 \\ x=-2 \end{gathered}[/tex]Final Answer
Therefore , the values of x and y are -2 and -3 respectively