The rule for the commutative property of addition is:
a + b = b + a
Stella writes:
[tex]-2\frac{1}{2}\text{ plus 3.7}[/tex]This can be expressed mathematically as:
[tex]-2\frac{1}{2}+3.7[/tex]Which according to the property of commutativity can also be written as:
[tex]\begin{gathered} 3.7\text{ + (-2}\frac{1}{2}) \\ 3.7\text{ - 2}\frac{1}{2} \end{gathered}[/tex]The expression Stella wrote based on the commutative property of addition is therefore:
[tex]\begin{gathered} -2\frac{1}{2}+\text{ 3.7 or} \\ 3.7\text{ - 2.5 (Note that 2}\frac{1}{2}=2.5) \end{gathered}[/tex]Solve for x and y.
8
5
Х
15
У
12
From the first diagram, we can write the equation:
[tex]x\text{ + y = 16 ----equation(1)}[/tex]From the second diagram, we can write the equation:
[tex]8\text{ + y = x ------equation (2)}[/tex]Substitute the expression for x in equation (2) into equation (1)
[tex]\begin{gathered} x\text{ + y = 16} \\ 8\text{ + y + y = 16} \\ 8\text{ + 2y = 16} \\ \text{Collect like terms} \\ 2y\text{ = 16 -8} \\ 2y\text{ = 8} \\ \text{Divide both sides by 2} \\ \frac{2y}{2}\text{ = }\frac{8}{2} \\ y\text{ = 4} \end{gathered}[/tex]Substituting the value of y into equation (2):
[tex]\begin{gathered} 8\text{ + y = x} \\ x\text{ = 8 + 4} \\ x\text{ = 12} \end{gathered}[/tex]Answer:
x = 12, y = 4
The perimeter of the rectangle below is 166 units. Find the length of side AB.Write your answer without variables.D4xС5x+2AB =0ХА
The working equation to find the perimeter of the rectangle is:
[tex]P=2L+2W[/tex]where L is the length of the rectangle and W is the width of the rectangle. Given the two sides of the rectangle and its parameter, we can rewrite the equation as:
[tex]166=2(4x)+2(5x+2)[/tex]Simplifying the equation and solve for x, we have:
[tex]\begin{gathered} 8x+10x+4=166 \\ 18x=166-4 \\ \frac{18x}{18}=\frac{162}{18} \\ x=9 \end{gathered}[/tex]For the side of AB that is the same as side AD, the measurement would be:
[tex]AB=5(9)+2=47[/tex]Answer: AB = 47
Exponential Growth Calculus need help
Answer:
(a) P(t) = 50·17.5^(t/1.5) or P(t) = 50·e^(1.9081t)
(b) P(5) ≈ 695,713
(c) P'(5) ≈ 1,327,514
(d) 4.5 hours
Step-by-step explanation:
You want the exponential function that describes bacterial growth from 50 cells to 875 cells in 1.5 hours, the population after 5 hours, and its rate of growth at that time. You also want to know the time at which the population reaches 250,000.
Exponential growthThe function that models exponential growth can be written as ...
population = (initial population)×(growth factor)^(t/period)
where (growth factor) is the population multiplier over the period.
(a) ExpressionHere, we are given an initial population of 50, and a growth factor of 875/50 = 17.5 in a period of 1.5 hours. This means we can write the function as ...
P(t) = 50(17.5^(t/1.5))
Note that the exact problem statement values are used, so no rounding is required.
If this is expressed using an exponent with a base of 'e', then we have ...
P(t) = 50e^(kt)
Comparing this to the above expression, we see ...
e^(kt) = 17.5^(t/1.5)
k = ln(17.5)/1.5 ≈ 1.9081 . . . . take natural logs and divide by t
So, ...
P(t) ≈ 50(e^(1.9081t))
(b) P(5)
The cell count after 5 hours is modeled as ...
P(5) = 50·e^(1.9081·5)
P(5) ≈ 695,713
(c) P'(5)Differentiating the population function, we have ...
P'(t) = 50·(1.9081)(e^(1.9081t) ≈ 95.4067e^1.9081t
Then the rate of change of the population at t=5 is ...
P'(5) = 1.9081·P(5)
P'(5) ≈ 1,327,514
(d) P^-1(250,000)The time required for the population to reach 250,000 can be found from ...
250,000 = 50·e^(1.9081t)
5,000 = e^(1.9081t) . . . . . . . divide by 50
ln(5,000) = 1.9081t . . . . . . . take natural logs
t = ln(5,000)/1.9081 . . . . . .divide by the coefficient of t
t ≈ 4.5
It will take about 4.5 hours for the population to reach 250,000.
The graph below shows the profits for a circus.a. What is the dependent variable?b. What is the independent variable?c. What is the slope (rate of change) of the line?d. What does this number mean for the circus?
Most of the time, the independent variable is located in the horizontal axis (x) and the dependent variable is located in the vertical axis (y), a change in the independent variable causes an influence on the value of the dependent variable, in this case, the dependent variable would be the profit in $ (thousands) and the independent variable is the # of tickets sold (thousands).
The slope of a line (m) is calculated by means of the formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Where (x1,y1) and (x2,y2) are points of the line, i this case, the x coordinate of the line represents # of tickets sold and the y coordinate the profit in $.
We can take whatever point from the line, I'll take the points (0,0) and (40,60), then the slope of the line is:
[tex]m=\frac{60-0}{40-0}=\frac{60}{40}=\frac{6}{4}=\frac{3}{2}[/tex]Then, the slope of the line equals 3/2
The slope of this line represents the earnings per ticket
a gardner has a cylindrical planter that has a diameter of 16 inches and is 2 feet deep. What is the approximate number od cubic feet of potting soil that he will need to fill the planter.
Given a cylinder with a diameter of d and a height of h, the volume, V, is given by
[tex]V=\frac{\pi d^2h}{4}[/tex]In this case,
d = 16 inches, h = 2ft
1 foot = 12 inches
Therefore,
[tex]d=\frac{16}{12}=\frac{4}{3}ft[/tex]Therefore,
[tex]V=\frac{\pi(\frac{4}{3})^22}{4}=\frac{\pi(4)(2)}{3^2}=\frac{8\pi}{9}\text{ cubic fe}et[/tex]V=2.79 cubic feet
Therefore the approximate number is 2.79
Which of the following is the once of a rhombus?
Product of its diagonals.
Half of sum of its diagonals
Half of product of its diagonals.
The answer is Half of product of its diagonals which is option C .
Let d1 and d2 be the diagonals of the rhombus.
A rhombus can be defined as a special parallelogram as it fulfills the requirements of a parallelogram, i.e. a quadrilateral with two pairs of parallel sides. In addition to this, a rhombus has all four sides equal just like a square. That is why it is also known as a tilted square.
Area of rhombus using diagonals (A) = 1/2 * d1 * d2.
where ,
d1 = length of diagonal 1
d2 = length of diagonal 2.
Therefore the answer is Half of product of its diagonals which is option C.
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Daniel's Print Shop purchased a new printer for $35,000. Each year it depreciates at a rate of 5%.Use an exponential function to find its approximate value after 8 years.
To solve, we will use the formula below;
V = V₀ ( 1- r)ⁿ
where
V₀ is the initial value
n is the number of years
r is the rate of time
From the question;
V₀= 35 000 r = 5/100 = 0.05 n =8
substitute the values into the formula and then evaluate
V = 35000 ( 1 - 0.05)⁸
= 35000(0.95)⁸
= $23219.72
help me pleaseeeeeeeeeeeeeeeeeeeeeee
thank you
Answer:
Domain: A, [tex][2, \infty)[/tex]
Range: [tex](-\infty, \infty)[/tex]
Step-by-step explanation:
The domain is the set of x-values and the range is the set of y-values.
point B divides pq in the ratio 1:3 if x cordinate is-1and the x corrdinate op is -3 what is the x cordinate of q
(VIEW THE IMAGE) Identify the step where the student made a mistake and explain the specific error that was made. Support your answer using the correct vocabulary. And how would you correct the mistake? This would be very helpful for me :)
The place where there was a mistake was in the second step of the equation.
What is an equation?It should be noted that an equation is used to show the relationship between the variables by equating them.
From the information, the equation is illustrated as:
-4(x - 15) + 5(-4x - 6y)
Multiply the values outside the parentheses by the values inside it.
= -4x + 60y -20x - 30y
Combine like terms
= -4x - 20x + 60y - 30y
= -24x + 30y
It should be noted that in the second step, In this place, the person didn't multiply the values well.
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I need help with this question. Not sure if i have to use sum formula or not
Answer
On the seventh day, 93,312 mails need to be sent.
Explanation
This question has a setup of a geometric progression.
The general form of a geometric progression is
aₙ = a rⁿ⁻¹
where
aₙ = nth term
a = first term = 128
r = common ratio = 3
n = number of days
So, for this question,
aₙ = a rⁿ⁻¹
aₙ = 128 (3ⁿ⁻¹)
On the first day, n = 1
aₙ = 128 (3ⁿ⁻¹)
a₁ = 128 (3¹⁻¹)
a₁ = 128 (3⁰) = 128 (1) = 128
On the seventh day, n = 7
aₙ = 128 (3ⁿ⁻¹)
a₇ = 128 (3⁷⁻¹)
a₇ = 128 (3⁶) = 128 (729) = 93,312
Hope this Helps!!!
17. Points W and Z are 8 unitsapart on the coordinate grid. Four facts aboutthe coordinates of the points are shown.• Point W is located at (-3, p).• The variable p has a value greater than 0.• Point Z is located at (-3, r).• The variable r has a value less than 0.What are possible values of p and r?p=OOlolOOO OCOOOOOOOOOOOOOSOOCOOOOOOOOOO0 0 0 0OOOOOOOCCO0 0
Possible values of p are from 1 to 7 while possible values of r are from -1 to -7
Here, we want to get the possible values of p and r
From the question, we have it that both W and Z are located at the same axis coordinate
What this mean is that it is a vertical line that connects both
Variable p has a value greater than zero means that it has a positive value
Variable r has a value less than zero means it has a negative value
Since the two are 8 units apart, we have it that;
p-r = 8
or p = 8 + r
Also, we have it that;
p > 0
r < 0
So, we need pairs of numbers that could work
We can use different values, minding the fact that p is positive and r is negative
If we have p = 7
r will be 7-8 = -1
p = 7 and r = -1 is a possible value
p = 6 and r = -2 is a possible value
p = 5 and r = -3 is a possible value
p = 4 and r = -4 is a possible value
p = 3 and r = -5 is a possible value
p = 2 and r = -6 is a possible value
p = 1 and r = -7 is a possible value
Thus, we have it that the value of p ranges from 1 to 7 while the value of r ranges from -1 to -7
Ava has 6 boxes of supplies for her classroom each box has a mass of 4 kg what is the total mass for all boxes
Given the mass of each box, and the total number of boxes, multiply both quantities to obtain the total mass, as shown below
[tex]6\cdot4=24\to6\text{ boxes, 4 }kg\text{ each}[/tex]Therefore, the answer is 24kg.
Reese is selling lemonade at the parade. He gets to keep 50% of the money he collects. A large lemonadeis $8.00 and a small lemonade is $2.00.The expression represents 50% of the money he collects,0.50(8/ + 2)Use the Distributive Property to expand the expression.The simplified expression is
Based on the given information, you have the following expression:
0.50(8l + 2s)
by using distributive property, you obtain:
0.50(8l + 2s) = 0.50(8l) + 050(2s)
= 4l + s
Hence, the simplified expression is 4l + s
if income tax expenses had been 23 % what would the net incime have been
EXPLANATION
The next income should be a 23% or 1.23 more the next month.
Janet cleans 25 of her bedroom that she shareswith her sister Mary. Mary also cleans 2/3 of theirshared bedroom. How much of the bedroom hasbeen cleaned by Janet and Mary?
Given
Janet = 25%
Mary = 2/3 = 66.66%
Procedure
Total = 25% + 66.66% = 91.66%
In fraction
11/12
Choose all the equations that have p = -(1/4) as the solution.A.) 3p + 2 = 5/4B.) 5p - 2 = 3/4 C.) 3p - 2 = 5/4 D.) 5p + 2 = 3/4
EXPLANATION
Given the equations in the problem, we just need to replace each p-term by -1/4 in order to see if this is a solution to each equation:
A.) 3*(-1/4) +2 = 5/4 = 5/4 --------> p = -(1/4) is a solution of this equation. ✔✔
B.) 5*(-1/4) - 2 = -13/4 ≠ 3/4 -------> p = -(1/4) is NOT a solution of this equation.✘✘
C.) 3*(-1/4) -2 = -11/4 ≠ 5/4 -------> p = -(1/4) is NOT a solution of this equation.✘✘
D.) 5*(-1/4) +2 = -11/4 ≠ 5/4 -------> p = -(1/4) is NOT a solution of this equation.✘✘
Help in example number two in the bottom right corner f(x) = -2x^2 - 8x + 1
First, find the vertex of the given function, we have
[tex]\begin{gathered} f(x)=-2x^2-8x+1 \\ \\ \text{The coefficients are } \\ a=-2,b=-8,c=1 \\ \\ \text{The x-coordinate of the vertex is at } \\ x=-\frac{b}{2a} \\ x=-\frac{-8}{2(-2)} \\ x=-\frac{-8}{-4} \\ x=-2 \end{gathered}[/tex]Next, substitute x = -2. to the given function and we get
[tex]\begin{gathered} f(x)=-2x^{2}-8x+1 \\ f(-2)=-2(-2)^2-8(-2)+1 \\ f(-2)=-2(4)+16+1 \\ f(-2)=-8+17 \\ f(-2)=9 \end{gathered}[/tex]Therefore, the vertex is at (-2.9).
The axis of symmetry is at x = -2.
The y-intercept at y = 1.
The x-intercepts are the following:
[tex]\begin{gathered} x=\frac{ -b \pm\sqrt{b^2 - 4ac}}{ 2a } \\ x = \frac{ -(-8) \pm \sqrt{(-8)^2 - 4(-2)(1)}}{ 2(-2) } \\ x=\frac{8\pm\sqrt{64-(-8)}}{-4} \\ x = \frac{ 8 \pm \sqrt{72}}{ -4 } \\ x = \frac{ 8 \pm 6\sqrt{2}\, }{ -4 } \\ \text{ Which becomes} \\ \\ x=\frac{8+6\sqrt{2}\,}{-4}\approx−4.12132 \\ x=\frac{8-6\sqrt{2}\,}{-4}\approx0.12132 \end{gathered}[/tex]Graphing the function we get
what is 28m-35 in factored form?
Answer:
[tex]28m-35=7\mleft(4m-5\mright)[/tex]Explanation:
Given the expression:
28m - 35
To write this in factored form, note that
[tex]\begin{gathered} 28=7\times4 \\ \text{and} \\ 35=7\times5 \end{gathered}[/tex]So
[tex]28m-35=(7\times4m)-(7\times5)[/tex]Because 7 is common to both 28m and 35, we factor out 7, to have
[tex]7\mleft(4m-5\mright)[/tex]A recipe calls for 3/4 cup sugar if the recipe is divided by 3 how much sugar is needed
Answer:
It'd be 1/4 a cup of sugar.
Step-by-step explanation:
Take 3/4 and divide it by 3/4 and because the denominator is the same, the top numbers divide each other and since 3 is divided by 3 is 1, you get the answer of 1/4 and then you can do that to all the other ingredients in the recipe.
Mayor has 20 prizes. He decided to give 1/2 to the relay game. How many of the prizes are for the relay games?
The total is 20 prizes. If he gives one half of them to the relay games, then the number of prizes for the relay games are 10 (it's 20 divided by 2)
Answer:
Step-by-step explanation:
10
one half of 20 is 10.
i’m stuck on this question. any help would be greatly appreciated
Using the Pythagorean theorem we get:
[tex]y^2+8^2=12^2.[/tex]Simplifying the above result we get:
[tex]y^2+64=144.[/tex]Subtracting 64 from the above equation we get:
[tex]\begin{gathered} y^2+64-64=144-64, \\ y^2=80. \end{gathered}[/tex]Therefore:
[tex]y=\sqrt{80}\approx8.9.[/tex]Answer:
[tex]y=8.9.[/tex]Can you prove these triangles are congruent from the information given?
Given:
Give names to the triangles
Aim:
We need to find whether these triangles are congruent.
Explanation:
The lines AE and DC are perbenticular.
[tex]\angle ABC\cong\angle\angle DBE\text{ since perpendicular lines makes right angles.}[/tex]Recall that the dashes on the lines show they are equal in length.
[tex]BC\cong BD\text{ Since both sides have single dashes.}[/tex][tex]AC\cong DE\text{ Since both sides have double dashes.}[/tex]Two sides and an angle not included between them are respectively equal to two sides and an angle of the other than the two triangles are equal.
[tex]\Delta ABC\cong\Delta DBE\text{ by SSA postulates}[/tex]Final answer:
The triangles ABC and DBE are congruent.
solve the formula for the specified variable
c=xy
solve for x
Answer: [tex]x= \frac{c}{y}[/tex]
Step-by-step explanation:
You want to isolate x on the right side ((x)=) This means making x alone.
To do so, you want to divide y on both sides. Division is the opposite of x.
[tex]\frac{c}{y}= \frac{xy}{y}[/tex]
The y's cancel out the right leaving:
[tex]\frac{c}{y} = x[/tex]
Evaluate the following expression for P=8000, r=9%, k=4, and n=16.
Given expression:
[tex]p(1\text{ + }\frac{r}{k})^{kn}[/tex]We have :
p = 8000
r = 9%
k = 4
n = 16
Substituting into the given expression:
[tex]=\text{ 8000}\times\text{ (1 + }\frac{0.09}{4})^{4\times16}[/tex]Evaluating:
[tex]\begin{gathered} =\text{ 8000 }\times(1+0.0225)^{64} \\ =\text{ 8000 }\times(1.0225)^{64} \\ =\text{ 33230.91155} \\ \approx\text{ 33230.91} \end{gathered}[/tex]Answer: 33230.91
A new long-life tire has a tread depth of 1/4 inch, instead of the more typical 5/32 inch. How much deeper is the tread on the new tire?The tread on the new tire is ___ inch deeper.(Simplify your answer. Type an integer or a fraction.)
To find how much deeper is the tread on the new tire, we just need to find the difference between the tread depth of this tire to the standard. To find this difference, we subtract the standard from the tread depth of this tire.
[tex]\frac{1}{4}-\frac{5}{32}[/tex]To realize this operation, we need both fractions to have the same denominator. If we multiply both numerator and denominator of a fraction by the same number, the fraction stays the same.
Since 32 is a multiple of 4, we can multiply the first fraction(both numerator and denominator) by 8 to put both fractions with the same denominator.
[tex]\frac{1}{4}\times\frac{8}{8}=\frac{8}{32}[/tex]Now, we can rewrite our operation
[tex]\frac{1}{4}-\frac{5}{32}=\frac{8}{32}-\frac{5}{32}[/tex]When we're doing an addition or subtraction between two fractions with the same denominator, we just need to do the operation on the numerator.
[tex]\frac{8}{32}-\frac{5}{32}=\frac{8-5}{32}=\frac{3}{32}[/tex]The tread on the new tire is 3/32 inch deeper.
A right triangle has one leg twice as long as the other and the perimeter is 18. Find the three sides of the triangle. Draw a diagram.
Let x represent the shortest leg of the triangle. If one leg twice as long as the other, then the length of the other leg is 2x.
Let y represent the hypotenuse which is the longest leg. The diagram is shown below
By applying pythagorean theorem,
y^2 = x^2 + (2x)^2
y^2 = x^2 + 4x^2 = 5x^2
Recall, perimeter of a triangle is the sum of the length of the sides of the triangle. If perimeter is 18, it means that
x + 2x + y = 18
3x + y = 18
y = 18 - 3x
Substituting y = 18 - 3x into y^2 = 5x^2, we have
(18 - 3x)^2 = 5x^2
324 - 54x - 54x + 9x^2 = 5x^2
9x^2 - 5x^2 - 54x - 54x + 324 = 0
4x^2 - 108x + 324 = 0
x^2 - 27x + 81 = 0
By applying the formula for quadratic equations, we have
[tex]\begin{gathered} x\text{ = }\frac{-\text{ b }\pm\sqrt[]{b^2-4ac}}{2a} \\ x\text{ = }\frac{-\text{ - 27}\pm\sqrt[]{-27^2-4(1\times81)}}{2\times1} \\ x\text{ = }\frac{27\pm\sqrt[]{405}}{2} \\ x\text{ = }\frac{27\text{ + 20.12}}{2}\text{ or x = }\frac{27\text{ - 20.12}}{2} \\ x\text{ = 23.56 or x = 3.44} \end{gathered}[/tex]The sides would be
3.44
2 x 3.44 = 6.88
18 - 3(3.44) = 7.68
The three sides are 3.44, 6.88 and 7.68
Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (-3,5) and parallel to x + 2y = 5.
we need to find the equation of a like, in two different ways
a) slope-interceppt Y=mX+b
b) standard form aX+bY=c
the information we have is,
it's parallel to x+2y=5 (this mean the have the same slope
so, let's find it
x+2y=5
2y=5-x
y=5/2 - 1/2 x
the slope is m= -1/2
The other thing we know is the line passing through (-3,5)
so when x=-3 the value of Y has to be 5
So let's check, until now we have the slope, our line is y= -1/2x + B
let's find out the value of B, replacing the values of x and y
5= -1/2 * -3 +B
5= 3/2 + B
7/2= B
So, the answer to (a) is=
Y= -1/2 X + 7/2
b) we need to change from slope to standard form
Y= -1/2 X + 7/2
Y + 1/2 X= 7/2
i think that's standard form, but also you can divide for 7/2 to get
2/7 Y + 1/7 X = 1
A community recently converted an old railroad corridor into a recreational trail. The graph at the right shows a map of the trail on a coordinate grid. The community plans to construct a path to connect the trail to a parking lot. The new path will be perpendicular to the recreational trail.A. Write an equation of the line representing the new path. B. What are the coordinates of the point at which the path will meet the recreational trail?
The equation of a line is represented by the equation y = mx+c where
m is the slope
c is the intercept
Slope of the line = y2-y1/x2-x1
slope of the line = 150-0/150-75
slope of te line = 150/75
slope of the line = 2
intercept is a point where the line cuts the y axis
The new path cuts the y axis at 100. Hence c = 100
The equation becomes y = 2x=100
2) The coordinates of the point where the path will meet recreational line is arounsd (80, 20) on the graph. THe line will cut the rereational line at this coordinates to make it perpendicular to the line recreation
largest 5-digit number rounded
to the nearest 100?
Answer:
100000
Step-by-step explanation:
the largest five digit number is 99999
to round it to the nearest hundred you must look at the tens place to see if it is above or below 5
it is above 5 so your round up
because it is a nine it becomes 10
this goes down the line until you get
100000
that is your answer
Answer:
99,999
it is the answer!
