EXPLANATION
Let's see the facts:
x= number of hot dogs they buy
y= number of drinks they buy
They buy 8 items, so
x + y = 8
Cost of a hot dog = $2
Cost of x hot dogs = 2x
Cost of a drink = $1
Cost of y drinks = 1y
The total amount they spent is:
2x + y
They spent $11, so
2x + y = 11
Now, we have a system of equations:
(1) x + y = 8
(2) 2x + y = 11
Subtracting (2) from (1):
2x + y = 11
- ( x + y = 8)
------------------
x = 3
Then, to find the number of drinks they bought, substitute x=3 into x + y = 8 and the solve for y:
3 + y = 8
Subtracting -3 to both sides:
3 - 3 + y = 8 -3
Simplifying:
y = 5
Answer:
They bought 3 hot dogs and 5 drinks.
If three times the sum of a number and 1 is 15, find the number.
The number is 4.6
Let the number be x
The equation will be 3x+1=15
on solving the equation we get
3x = 15-1
X = 14/3
X = 4.6
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Graph: x < -2 I need help graphing this problem
Kindly check below.
1) Usually, whenever we need to graph an inequality, we need to treat it as if it was an equation.
2) So if we consider that x=-2 is a vertical line that passes through point -2, we can start with that.
3) Since the sign is < then we need to plot a dashed line delimiting the region to be shaded, given that -2 is not included. And finally, as it is lesser than -2 we can paint the region to the left of -2, where the numbers lesser than -2 are located.
4) So, our graph is:
To estimate the height of a building, two students find the angle of elevation from a point (at ground levedown the street from the building to the top of the building is 30°. From a point that is 400 feet closer tothe building, the angle of elevation (at ground level) to the top of the building is 52°. If we assume thatthe street is level, use this information to estimate the height of the building.The height of the building isfeet.
Given:
ground to building top is 30 degrees.
Distance = 400 feet
The angle of elevation at is top of the building = 52 degrees.
Find-: Height of the building.
Sol:
For the triangle ABC
Perpendicular = Height
Base = x
Angle = 52
Use trigonometric formula:
[tex]\begin{gathered} \tan\theta=\frac{\text{ Perpendicular}}{\text{ Base}} \\ \\ \tan52=\frac{H}{x} \\ \\ 1.2799=\frac{H}{x} \\ \\ x=\frac{H}{1.2799} \end{gathered}[/tex]For the triangle ABD is:
Perpendicular = Height
Base = x+400
Angle = 32
[tex]\begin{gathered} \tan\theta=\frac{\text{ Perpendicular}}{\text{ Base}} \\ \\ \tan32=\frac{H}{x+400} \\ \\ 0.6249=\frac{H}{x+400} \\ \\ \end{gathered}[/tex]Put the value of "x" is:
[tex]\begin{gathered} 0.6249(x+400)=H \\ \\ 0.6249x+249.947=H \\ \\ 0.6249(\frac{H}{1.2799})+249.947=H \\ \\ 0.488H+249.947=H \\ \\ 0.512H=249.947 \\ \\ H=488.407 \\ \end{gathered}[/tex]So the height of the building is: 488.407 feet
Pick a situation related to your major that would be modeled by one of the models discussed in class. Describe why this model would be helpful. Models are any application/word problem used for exponential/logarithmic functions. For example, exponential growth or decay are models of exponential functions. Now, choose one that would fit with your career path and describe why it would be helpful.
To measure sound, we use the measure decibel which in fact is a logartimic funcion of the amplitude of a soud
To measure eartquakes, we use the Richter scale whci is a logaritmic function of the amplitude of the force of an earthquake
To calculate how long a body has been dead, the coroner must know how long the body temperature has not been at 98.6 degrees. Because the rate of the body cooling is proportionate to temperature differences between the body and its surroundings, the answer is found by calculating exponential decay using logarithms.
Find the product of the complex numbers. Leave your answer in polar form
Given: Two complex numbers below
[tex]\begin{gathered} z_1=2+2i \\ z_2=-3+3i \end{gathered}[/tex]To Determine: The product of the given complex numbers
[tex]z_1z_2=(2+2i)(-3+3i)[/tex][tex]\begin{gathered} z_1z_2=2(-3+3i)+2i(-3+3i) \\ z_1z_2=-6+6i-6i+6i^2 \\ z_1z_2=-6+6i^2 \end{gathered}[/tex]Please note that
[tex]\begin{gathered} i=\sqrt[]{-1} \\ i^2=(\sqrt[]{-1})^2_{} \\ i^2=-1 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} z_1z_2=-6+6i \\ z_1z_2=-6+6(-1) \\ z_1z_2=-6-6 \\ z_1z_2=-12+0i \end{gathered}[/tex]Let us convert the product to polar form
Please note that
[tex]\begin{gathered} if,z=x+iy,the\text{ polar form is} \\ z=r(\cos \theta+i\sin \theta) \\ \text{where} \\ r=\sqrt[]{x^2+y^2} \\ \tan \theta=\frac{y}{x} \\ \theta=tan^{-1}(\frac{y}{x}) \end{gathered}[/tex]Apply the conversion into the product we got
[tex]\begin{gathered} z_1z_2=-12+0i,x=-12,y=0 \\ r=\sqrt[]{x^2+y^2}=\sqrt[]{(-12)^2+0^2} \\ r=\sqrt[]{144+0} \\ r=\sqrt[]{144} \\ r=12 \\ \theta=\tan ^{-1}(\frac{0}{-12}) \\ \theta=\tan ^{-1}(0) \\ \theta=\pi \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} z_1z_2=r(\cos \theta+i\sin \theta) \\ r=12,\theta=\pi \\ z_1z_2=12(\cos \pi+i\sin \pi) \end{gathered}[/tex]Hence, the product of the complex numbers in polar form is
12(cosπ+isinπ)
2x+3y=5 at (-2,3) find the equation of the tangent line
The slope of the tangent line to the line 2x+3y=5 can be found by differentiating 2x+3y=5.
Differentiating 2x+3y=5 with respect to x, we get
[tex]\begin{gathered} 2+3\frac{dy}{dx}=0 \\ 3\frac{dy}{dx}=-2 \\ \frac{dy}{dx}=\frac{-2}{3} \end{gathered}[/tex]m=dy/dx is the slope of tangent line.
Hence, slope, m=-2/3.
Now, the equation of the tangent line passing through point (x1, y1)=(-2, 3) with slope m=-2/3 can be found as,
[tex]\begin{gathered} m=\frac{y_1-y}{x_1-x} \\ \frac{-2}{3}=\frac{3-y}{-2-x} \\ -2(-2-x)=3(3-y) \\ 4+2x=9-3y \\ 3y+2x=5 \end{gathered}[/tex]Therefore, the equation of the tangent line is 3y+2x=5.
A zebra and a giraffe are having a race, 300 yards from a row of trees (the starting line) to the edge of a stream (the finish line). When they start evenly, the zebra wins the race by 50 yards.They decide to race again, but in the second race the zebra has to start 50 yards behind the row of trees (350 yards from the finish line), while the giraffe starts at the usual starting line. The zebra and the giraffe always run at the same speeds from race to race. Who wins the second race? Explain your reasoning.
A zebra and a giraffe are having a race, 300 yards from a row of trees (the starting line) to the edge of a stream (the finish line). When they start evenly, the zebra wins the race by 50 yards.
They decide to race again, but in the second race the zebra has to start 50 yards behind the row of trees (350 yards from the finish line), while the giraffe starts at the usual starting line. The zebra and the giraffe always run at the same speeds from race to race. Who wins the second race? Explain your reasoning.
we have that
The speed is equal to divide tthe distance by the time
speed=d/t
t=d/speed
The zebra and the giraffe always run at the same speeds from race to race.
so
Race 1
Zebra ------> t=300/speed1
giraffe ----> t=(300-50)/speed2 -----> t=250/speed2
where t is the finishing time of the zebra
equate both equations
300/speed1=250/speed2
300/250=speed1/speed2
1.2=speed1/speed2
speed1=1.2speed2
that means ----> the spped of the zebra is 1.2 times the speed of the giraffe
Race 2
Zebra ----> t1=350/(1.2speed2) ------> t1=292/speed2
Giraffe ----> t2=300/speed2 ------> t2=300/speed2
compare the times
t1 < t2
that means
the zebra wins the race 25. The table shows the amount of money, A, in a savings account after mmonths. Select ALL the equations that represent the relationship betweenthe amount of money, A, and the number of months, m.*number ofmonthsdollaramount51,20061,30071,40081,500
Give the range of the relation. 2), (18, 4) (11, -3), (2, -2), (2, 0), (6, a. range: 11, 6, 2, 18 b. range: -3, -2, 2, 4 c. range: 11, 6, 0, 2, 18 d. range: -3, -2, 0, 2, 4
Give the range of the relation. (18, 4) (11, -3), (2, -2), (2, 0), (6, 2)
a. range: 11, 6, 2, 18
b. range: -3, -2, 2, 4
c. range: 11, 6, 0, 2, 18
d. range: -3, -2, 0, 2, 4
we know that
the range are the possible values of y in the data set
so
In this problem
range 4,-3,-2,0,2
therefore
the answer is the option dExampleIf you have(3,1), (2,8), (7,3), (5,0)the range are the values of ysothe value of y is the second coordinate in each ordered pair(3,1) ------> y-coordinate is 1(2,8)-----> 8(7,3) ----> 3(5,0) ----> 0thereforethe range is 1,8,3,0orderRange (0,1,3,8)The mid point between T(-2,6) and J (-5,1)
To find the midpoint between two points wee need to apply the following formula:
[tex]\begin{gathered} x_m=\frac{x_1+x_2}{2} \\ y_m\text{ = }\frac{y_1+y_2}{2} \end{gathered}[/tex]Where (x1,y1) are the coordinates of the first point, (x2,y2) are the coordinates of the second point and (xm,ym) are the coordinates of the mid point. In this case the first point is T and the second point is J. If we apply the formula we should find the coordinates of the midpoint between them.
[tex]\begin{gathered} x_m\text{ = }\frac{-2-5}{2} \\ x_m\text{ = }\frac{-7}{2} \\ x_m\text{ = -3.5} \end{gathered}[/tex][tex]\begin{gathered} y_m\text{ = }\frac{6+1}{2} \\ y_m=\frac{7}{2} \\ y_m\text{ = 3.5} \end{gathered}[/tex]The coordinates of the midpoint are (-3.5, 3.5).
use synthetic division to find the awnser write the quotient and write any remainders in fraction form (3x^3-x^2-9x+5)÷(x-2)
The given expressions are
[tex]\begin{gathered} 3x^3-x^2-9x+5 \\ x-2 \end{gathered}[/tex]We have to use the coefficients only to divide. Remember that the number 2 must multiply all the numbers in the quotient. The synthetic division would be:
x3 x2 x
| 3 -1 -9 5
2 | 6 10 2
3 5 1 7
Notice that the remainder is 7, the quotient would be the coefficients under the line.
The quotient is[tex]3x^2+5x^{}+1[/tex]Notice that the quotient doesn't include the number 7 because that's the remainder.
The remainder is 7.We can express the quotient, the remainder, and the divisor as follows:[tex]\frac{3x^3-x^2-9x+5}{x-2}=3x^2+5x+1+\frac{7}{x-2}[/tex]Sallys recipe for chocolate chip cookies yields 48, 1 oz cookies. If she want to make 48,2 oz cookies what is her conversation factor??(Hint : how many total oz cookies is in the original recipe yield and the new recipe yields)
Tano went to a professional basketball game He spent $54.85 for a ticket S4 99 for a hot dog and $2 15 for a soda What is the total amount that he spent? Use mental math to find the sum. A S60.99 O B 961.00 O C 561.99 OD 96200
The total amount is calculated as the sum of the cost of the ticket, the cost of the hot dog, and the cost of the soda.
So, the total amount is:
[tex]54.85+4.99+2.15=61.99[/tex]Answer: $ 61.99
To answer it using mental math, we can add 0.85 and 0.15 and get 1, add 54 and 2 and get 56 and approximate 4.99 to 5.
Then, the sum 1 + 56 + 5 is equal to 62.00 but we need to subtract 0.01 because of the approximation. Therefore the final result is:
62.00 - 0.01 = 61.99
Give the domain of y = ln x.
Remember that
[tex]y=\ln x=\log _ex\Leftrightarrow e^y=x[/tex]And e^y can have any value in the interval (0, infinite).
Therefore, the domain of ln(x) is
[tex]\text{domain(}\ln (x))=(0,\infty)[/tex]The answer is x in (0, infinite)
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.Carter and Anna are making presentations for a class project. Carter's slideshow starts with a verbal introduction that is 13 seconds long, and then each slide is left up for 10 seconds. Anna leaves each slide onscreen for 6 seconds, and her introduction lasts 17 seconds. Carter and Anna notice that their presentations have both the same number of slides and the same duration. How long is each presentation? How many slides are in each presentation?
Given
Carter and Anna are making presentations for a class project.
Carter's slideshow starts with a verbal introduction that is 13 seconds long, and then each slide is left up for 10 seconds.
Anna leaves each slide onscreen for 6 seconds, and her introduction lasts 17 seconds.
Carter and Anna notice that their presentations have both the same number of slides and the same duration.
To find how long is each presentation and how many slides are in each presentation.
Now,
Let x be the time taken for each presentation and y be the number of slides.
Then, the equations are,
[tex]\begin{gathered} 13+10y=x\text{ \_\_\_\_(1)} \\ 17+6y=x\text{ \_\_\_\_\_(2)} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} 13+10y=17+6y \\ 10y-6y=17-13 \\ 4y=4 \\ y=1 \end{gathered}[/tex]Substitute y=1 in (1).
Then,
[tex]\begin{gathered} 13+10y=x \\ 13+10\times1=x \\ x=13+10 \\ x=23 \end{gathered}[/tex]Hence, the time taken for each presentation is 23 seconds and the number of slides is 1.
Question 3 The graph of the equation x + 3y=6 intersects the y- axis at which coordinate point? (0, 2) (0, 6) (0, 18) (6, 0)
The intersection with the y-axis ocurrs at x=0. That is, by substituting x=0 in our given expression, we get
[tex]0+3y=6[/tex]which gives
[tex]\begin{gathered} 3y=6 \\ y=\frac{6}{3} \\ y=2 \end{gathered}[/tex]Then, the intersection point is (0,2). Which corresponds to the first option.
The ratio of men to women in a certain factory is 3 to 4. There are 228 men. find the number of women
Given
The ratio of men to women in a certain factory is 3 to 4. There are 228 men. find the number of women
Solution
Let the number of women be x
[tex]\begin{gathered} Men\text{ : Women} \\ 3\text{ : 4} \\ 228:\text{ x} \end{gathered}[/tex][tex]\frac{3}{4}=\frac{228}{x}[/tex]cross multiply
[tex]\begin{gathered} 3\times x=4\times228 \\ 3x=912 \end{gathered}[/tex]Divide both sides by 3
[tex]\begin{gathered} \frac{3x}{3}=\frac{912}{3} \\ \\ x=304 \end{gathered}[/tex]The number of women is 304
Which number is the solution of n/3 = - 12
-36 is the solution to the given equation
Here, we want to find the value of n that solves the equation
We simply cross-multiply here
We have;
[tex]n\text{ = 3 }\times\text{ -12 = -36}[/tex]Answer:
Step-by-step explanation:
1. The answer is -36 because -36/3=12.
please help me figure out how to determine the range of the following graph (the line y=5 is a horizontal asymptote)
we are given the graph of the function and we are interested in finding the range of the function. Recall that the range of a graph is simply the set of values on the y axis, for which there is a point on the graph that has that y coordinate.
One easy way to spot this set, is by taking any point on the graph and then drawing a horizontal line. Wherever the line crosses the y axis, that point is included in the range.
From the graph, we can see that no part of the graph has values with y coordinate less than 5. That is, any number less than 5 in the y coordinate would indicate that there is no point on the graph at that "height". So every number less than 5 is excluded from the range.
We are also told that line y=5 is a horizontal asymptote. This means that despite the graph is really close to the line y=5 (and it keeps getting closer and closer as x increases), it never touches the line. This means that the point 5 is excluded from the range.
Finally, we can see that above the horizontal line y=5, if we draw a horizontal line on the graph, it will touch the y axis. This means that every number greater than 5 is part of the range. Then, the set of numbers that represent the range is
[tex](5,\text{infinity)}[/tex]which of the following sets of numbers could represent the three sides of a triangle. 4,7,1212,16,284,12,137,13,22
ANSWER
4, 12, 13
EXPLANATION
The sum of any 2 sides of a triangle must be greater than the measure of the third side.
For these options we have to check which set of numbers follow this rule:
[tex]\begin{gathered} 4+7>12 \\ 11>12\text{ }\rightarrow\text{ false} \end{gathered}[/tex][tex]\begin{gathered} 12+16>28 \\ 28>28\text{ }\rightarrow\text{ false} \end{gathered}[/tex][tex]\begin{gathered} 4+12>13 \\ 16>13\text{ }\rightarrow\text{ true} \\ 12+13>4 \\ 25>4\text{ }\rightarrow\text{ true} \\ 4+13>12 \\ 17>12\text{ }\rightarrow\text{ true} \end{gathered}[/tex][tex]\begin{gathered} 7+13>22 \\ 20>22\rightarrow\text{ false} \end{gathered}[/tex]The third set of numbers could represent the three sides of a triangle
In an all boys school, the heights of the student body are normally distributed with a mean of 71 inches and a standard deviation of 3.5 inches. Out of the 1707 boys who go to that school, how many would be expected to be taller than 75 inches tall, to the nearest whole number?
The formula for the z score of a number is given by:
[tex]z=\frac{x-\overline{x}}{\sigma}[/tex]Where:
[tex]\begin{gathered} x=\text{ the observed value} \\ \overline{x}=\text{ the mean} \\ \sigma=\text{ the standard deviation} \end{gathered}[/tex]In this case,
[tex]\begin{gathered} x=75 \\ \overline{x}=71 \\ \sigma=\text{ 3.5} \end{gathered}[/tex]Therefore, the z score of x=75 is given by:
[tex]z=\frac{75-71}{3.5}=\frac{4}{3.5}\approx1.143[/tex]Therefore, the probability that a boy is taller than 75 inches is given by the area under the normal probability distribution curve between z=1.143 and z=∞, P(z > 1.143):
The area is approximately 0.1265.
Therefore, the required probability is 0.1265.
Convert the probability to percent by multiplying with 100:
[tex]0.1265\times100=12.65[/tex]Hence, about 12.65 % of all the boys are taller than 75 inches.
Therefore, the total number of boys that are taller than 75 inches is given by:
[tex]\frac{12.65}{100}\times1707\approx216[/tex]Therefore, the number of boys expected to be taller than 75 inches is approximately:
216
Use the Pythagorean theorem to find each missing length to the nearest tenth I want you tell how to do please
Let the unknown side be h
Pythagoras theorem states that the square of the hypotenuse is equal to the square of the opposite plus the square of the adjacent in any right angled triangle
[tex]\begin{gathered} \text{From the figure, } \\ \text{hypotenus = h} \\ \text{Let opposite = 3, adjacent = 10} \\ \text{Therefore, applying pythagoras theorem,} \\ h^2=3^2+10^2 \\ h^2=9+100 \\ h^2=109 \\ h=\sqrt[]{109} \\ h=10.44 \\ h\approx10.4\text{ (nearest tenth)} \end{gathered}[/tex]The missing length is 10.4
Determine the number of significant figures in the measurement 77.09 m.Express your answer numerically as an integer.
Given:
The significant figures in 77.09 m are 4 because all the digits are necessary to denote the quantity.
Hence, 4 is the number of significant figures expressed in integer.
I need help with this I need to know what I’m doing wrong.. do I need to put a negative for (y+8^2) or a positive 8 so confused… help #4write in standard equation for a circle and identify center and radius
4) You have the following equation:
[tex]x^2+10x+y^2-16=0[/tex]In order to determine the radius and center of the circle, complete squares for x. You don't complete squares for y because there is no term with y in the given expression. It is only a y^2 term.
By adding 25 and subtracting 25 left side of the equation you obtain:
[tex]x^2+10x+25+y^2-16-25=0[/tex]The first three terms are a perfect square (x + 5)^2, then, by using this factor and by simplifying in the previous equation you can write:
[tex](x+5)^2+y^2-41=0[/tex]Finally, add 41 both sides:
[tex](x+5)^2+y^2=41[/tex]The previous equation is in standard form for a circle equation:
[tex](x-h)^2+(y-k)^2=r^2[/tex](h,k) is the center of the circle and r the radius. By comparin the previous equation with the expression you obtain you obtain:
center of the circle = (-5,0)
radius r = √41
Las 25 22. Lines a and bare parallel and cut by the transversalt. Label each of the 7 unknown angles with the correct angle measure. 20 t
Explanation:
Angle b and the one that's 70º are vertical angles, so they are congruent.
Angles b and g are corresponding angles. They are congruent.
Angles g and e are vertical angles. They are congruent.
Angles a and b are supplementary - their measures add up 180º:
[tex]\begin{gathered} m\angle a+m\angle b=180º \\ m\angle a=180º-70º \\ m\angle a=110º \end{gathered}[/tex]Angles a and d are corresponding angles. They are congruent.
Angles c and a are vertical angles. They are congruent.
Angles d and f are vertical angles. They are congruent.
Answers:
• m∠a = 110º
,• m∠b = 70º
,• m∠c = 110º
,• m∠d = 110º
,• m∠e = 70º
,• m∠f = 110º
,• m∠g = 70º
write each degree measure in radians round to the nearest hundred 76 degrees 124 degrees and 149 degrees
Conversion from Degrees to Radians
One radian is equivalent to 180°
To convert from degrees to radians, we just multiply by the factor:
[tex]\frac{\pi}{180}[/tex]a) Convert 76 degrees to radians
[tex]76\cdot\frac{\pi}{180}=1.33\text{ rad}[/tex]b) Convert 124 degrees to radians
[tex]124\cdot\frac{\pi}{180}=2.16\text{ rad}[/tex]c) Convert 149 degrees to radians
[tex]149\cdot\frac{\pi}{180}=2.60\text{ rad}[/tex]Z1 and Z2 are a linear pair and the mZ1 is 9 times the measure of Z2. Find mZ1.mZ1 =degrees
You bought a notebook and four erasers at Target. The notebook cost 5$.You spent a total of 25$ at target. How much did each eraser cost
You bought:
1 notebook
4 eraser
Total Cost = 25.
Now, given Notebook = $5, so
25 - 5 = 20 dollars are left for [4 erasers]
So, each eraser would cost:
20/4 = 6 dollars
Thus,
Each Eraser Cost = $6
Which two parallelograms have the same area? Figure 1 Figure 2 18 cm 4 cm 13 cm 1 1 15 Figure 4 cm Figure 3 11 cm 5 cm 6 cm 1 112
Figure 1 Area = 4 x 15 = 60 cm²
Figure 2 Area = 3 x 18 = 54 cm²
Figure 3 Area = 6 x 11 = 66 cm²
Figure 4 Area = 12 x 5 = 60 cm²
Answer:
Therefore area of figure 1 and 4 are equal
the following data set shows the number of books checked out from the library during the first two weeks of the month .36, 39, 40, 42, 45, 2, 38, 41, 37, 38, 35, 37, 35, 38
According to the given data set, we can observe that there's one day where people only checked out 2 books which is way too low compared to the other days, which represents an outlier.
Hence, the answer is 1) there is one outlier, indicating very few books were checked out on that day.