Use the graphs of f and g to solve Exercises 87, 88, and 89.
87. Find the domain of f + g.
88. Find the domain of [tex]\frac{f}{g}[/tex].
89. Graph f + g.
(You can just explain how to graph it for #89.)
Answers
Answers and explanations:
87. The domain of added functions includes the restrictions of both. So the range of the added function in this question is [-4, 3]
88. When finding the domain of a divided function we do the same as adding, but with an extra rule: g can't equal zero. So for this question the domain is (-4, 3)
89. To graph f + g you add the y-values for each x-value. I added a picture to help explain this one!
Answer:
87. [-4, 3]
88. (-4, 3)
89. See Attachment
General Formulas and Concepts:
Algebra I
Reading a Cartesian PlaneCoordinates (x, y)FunctionsFunction NotationDomains - the set of x-values that can be inputted into a function f(x)[Interval Notation] - Brackets are inclusive, (Parenthesis) are exclusiveStep-by-step explanation:
*Notes:
When adding functions, the domain of the new function is defined as the intersections of the domains of f and gWhen dividing functions, the domain of the new function is defined as the intersections of the domains of f and g except for the points where g(x) = 0 (this is because we cannot divide by 0)Step 1: Define
Identify the domains of each function.
Domain of f(x): [-4, 3]
Domain of g(x): [-5, 5]
Step 2: Find 87.
Determine the x-values for each function that overlap/intersect.
f(x) and g(x) have intersect from -4 to 3.
Domain f + g: [-4, 3]
Step 3: Find 88.
Determine the x-values for which g(x) = 0.
The function g(x) equals 0 at x = -4 and x = 3. Therefore, these x-values are excluded in the domain.
Domain of f/g: (-4, 3)
Step 4: Find 89.
To draw a graph of the f + g, we must combine the y-values for each x-value domain in a t-chart and plot by hand.
x | f(x) x | g(x) x | f + g
-4 5 -4 0 -4 5
-3 4 -3 1 -3 5
-2 3 -2 2 -2 5
-1 3 -1 2 -1 5
0 2 0 1 0 3
1 1 1 1 1 2
2 -1 2 1 2 0
3 -3 3 0 3 -3
Related Questions
A cash register at a store contains $66 bills. There are 6 more $5 bills than $10 bills. The number of $1 dollar bills is three times more the number of $10 dollar bills. How many bills of each kind are there?
Answers
Answer: There are 8 $5 bills, 2 $10 bills, and 6 $1 bills.
Step-by-step explanation: If you have 6 more $5 bills than the number of $10 bills, that means you have at least 6 to start off with. This also means you have at least 1 $10 bill. If you have 1 $10 bill, then you have 3 $1 bills as well. Adding those bills up, you get $43. From here, you know you need $3 more. Meaning you have to add another $10 bill. Since you added another $10 bill, you have to equal out the $5 bills.
So if you know you have 2 $10 bills and 6 $1 bills, you can determine that you need 8 $5 bills to complete the set.
2 $10 = $20
8 $5 = $40
6 $1 = $6
$20 + $40 + $6 = $66
Kim made 1 1/4 quarts of a fruit smoothie. She drank 1/5 of her smoothie. Her brothers drank the rest. They each had 1/3 quart. How many brothers does Kim have?
Answers
Answer:
3 brothers.
Step-by-step explanation:
You know that Kim made [tex]1\frac{1}{4}[/tex] quarts of a fruit smoothie.
Observation: [tex]1\frac{1}{4}[/tex] is the same as saying [tex]\frac{5}{4}[/tex] because, [tex]1\frac{1}{4} =1+\frac{1}{4} =\frac{4.1+1}{4} =\frac{5}{4}[/tex], then Kim made [tex]\frac{5}{4}[/tex] of a fruit smoothie.
Now, the problem says that Kim drank [tex]\frac{1}{5}[/tex] of her smoothie, this means:
[tex]\frac{5}{4} .\frac{1}{5}=\frac{1}{4}[/tex]
Kim drank [tex]\frac{1}{4}[/tex] quart of the smoothie, the rest of the smoothie is:
[tex]\frac{5}{4}- \frac{1}{4}=\frac{4}{4}[/tex]
Now to know how many brothers Kim has we have to divide the rest of the smoothie ([tex]\frac{4}{4}[/tex]) in [tex]\frac{1}{3}[/tex], this is:
[tex]\frac{4}{4} :\frac{1}{3} =\frac{12}{4} =3[/tex]
Then Kim has 3 brothers.
Addison painted her room. She had 505050 square meters to paint, and she painted at a constant rate. After 222 hours of painting, she had 353535 square meters left. Let yyy represent the area (in square meters) left to paint after xxx hours. Complete the equation for the relationship between the area and number of hours.
Answers
Answer:
y = 50 - 7.5x
Step-by-step explanation:
Given,
The original area for painting = 50 m²,
Let c meter per hour be the constant rate of painting,
So, after 2 hours, the area painted = 2c m²,
Thus, the area left to paint = 50 - 2c
According to the question,
50 - 2c = 35
2c = 15
⇒ c = 7.5
i.e. the constant rate of painting is 7.5 meters per hour,
Hence, the area left after x hours = 50 - 7.5x
If y represents the area left to paint after x hours,
Then,
y = 50 - 7.5x
Which is the required equation.
A company had the following direct materials cost information:
Standard costs assigned: Direct materials standard cost (405,000 units @ $2/unit) $810,000
Actual costs: Direct Materials costs incurred (403,750 units @ $2.20/unit) $888,250
What was the cost variance?
a) $2,500 Favorable.
b) $78,250 Favorable
c) $78.250 Unfavorable
d) $80,750 Favorable.
e) $80,750 Unfavorable.
Answers
Answer:
Option c) $78,250 unfavorable
Step-by-step explanation:
The given information in the question :
Standard cost = $810,000
Actual cost = $888,250
Cost variance can be calculated as using the following formula :
Cost variance = standard cost - actual cost
= 810,000 - 888,250
= ($78,250) unfavorable
Favorable indicates how much under budget the project.
unfavorable indicates how much over budget the project.
Therefore, Option c) 78,250 unfavorable is the answer.
Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 800 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 8 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value. (Round your answer to one decimal place.)
Answers
Answer:
t = 100 ln 100
Step-by-step explanation:
D(t) : The amount of dye (in g) at time t (in min)
D(0) = 800 L * 1 g/L = 800 g
the change in D is:
[tex]\frac{dD(t)}{dt} =D_{in}- D_{out} \\D_{in}: 0*8\ g/min \\D_{out}: \frac{D(t)}{800} *8\ g/min \\\frac{dD(t)}{dt} = -\frac{1}{100}D(t)[/tex]
[tex]\frac{dD(t)}{D(t)} =-\frac{1}{100}dt \\\int\limits^{D(t)}_{800} {\frac{1}{D(t)} } \, dD(t) =\int\limits^t_0 {t} \, dt \\ln(\frac{D(t)}{800})=-\frac{1}{100}t \\D(t) = 800e^{-\frac{1}{100}t} \\Solving\ D(t) = 0.01* D(0)=0.01*800 =8 \\8 = 800e^{-\frac{1}{100}t} \\ln (\frac{1}{100})=-\frac{1}{100}t \\100 ln 100 = t[/tex]
An amusement park offers a yearly membership of $275 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12 and 15 visits to the park. c. Plot these points on a coordinate graph and describe things you notice from the graph.
Answers
Answer: (6+15+9)=275+5x
X=11
Step-by-step explanation:
Supposing one car by visitor, then non members will always pay the 5 dollars per person parking
Non members will spend 30 dollars a visit
And members wil have an accumulated spend of 275 initial dollars plus 5 dollars a visit.
Then (6+15+9)=275+5x
X=11
They'll have spent the same after the 11th visit.
Chart and plot in picture.
A seven digit numer that has a 0 in the ones place a 6 in the ten thousends place an 8 in the millions place and fives in each of the remaining places.What is the number
Answers
That number is 8,565,550.
Answer:
8,565,550.
Step-by-step explanation:
The number is 8,x6x,xx0 where all the x's = 5 so the answer is:
8,565,550.
Mrs. Canon and Mrs. Solace are both getting their nails done today. Mrs. Canon gets her nails done every 8 days. Mrs. Solace gets her nails done every 12 days. In how many days will they be at the nail salon on the same day again?
Answers
Answer:
24 days
Step-by-step explanation:
The least common multiple (LCM) of 8 and 12 is 8·3 = 12·2 = 24.
The ladies will be at the nail salon on the same day again in 24 days.
_____
8 = 2³
12 = 2²·3
The LCM will have these factors to their highest powers: 2³·3 = 24.
__
The LCM is also the product divided by their greatest common factor (GCF). GCF(8, 12) = 4, so ...
LCM(8, 12) = 8·12/4 = 24
Answer:
24
Step-by-step explanation:(LCM) 8,16,24
(LCM)12,24
LCM is 24 so the answer to the question is 24
Based on an 8-hour day, the number of hours worked in a hospital food service department was 55,267/yr, and the total number of hours paid was 59,995/yr. The actual number of productive FTEs was:
a. 2.27b. 18.90c. 26.60d. 28.80
Answers
Final answer:
To find the actual number of productive FTEs, divide the total number of hours worked (55,267) by the number of hours worked per FTE. The answer is option a. 2.27.
Explanation:
To find the actual number of productive FTEs, we need to divide the total number of hours worked (55,267) by the number of hours worked per FTE. The number of hours worked per FTE can be calculated by dividing the total number of hours paid by the total number of productive FTEs. So, the equation becomes:
55,267 / (59,995 / x) = x
Multiplying both sides of the equation by (59,995 / x), we get:
55,267 = (59,995 / x) * x
Simplifying further:
55,267 = 59,995
Dividing both sides of the equation by 59,995, we get:
x = 55,267 / 59,995
x = 0.9213
Therefore, the actual number of productive FTEs is approximately 0.9213, which can be rounded to 0.92. Therefore, the answer is option a. 2.27.
Final answer:
The actual number of productive Full-Time Equivalents (FTEs) is calculated by dividing the total annual productive hours (55,267 hours/year) by the standard annual working hours for one full-time employee (2,080 hours/year), resulting in 26.57, which rounds to option c. 26.60.
Explanation:
To calculate the actual number of productive Full-Time Equivalents (FTEs) based on the hours worked in a hospital food service department, we use the given number of hours worked per year and divide it by the standard number of working hours in a year for one full-time employee.
First, let's establish the standard number of working hours in a year for one FTE, based on an 8-hour day:
1 workday = 8 hours1 workweek = 5 workdays (typically for full-time)1 workyear (excluding holidays/vacations) = 52 workweeksTotal working hours in a year = 8 hours/day × 5 days/week × 52 weeks/year = 2,080 hours/yearTo find the actual number of productive FTEs, we divide the number of hours worked by the standard number of working hours in a year:
Productive Hours Worked: 55,267 hours/year
Standard Hours for 1 FTE: 2,080 hours/year
Actual number of productive FTEs = Productive Hours Worked / Standard Hours for 1 FTE
Actual number of productive FTEs = 55,267 hours/year / 2,080 hours/year
Actual number of productive FTEs = 26.57
Therefore, the nearest option to our result is c. 26.60.
The binomial coefficient Subscript n Baseline Upper C Subscript x gives the number of ways of picking a subset of x items out of n. Using this fact, how many ways are there to pick a committee of 4 from among all of the office employees?
Answers
Answer:
[tex]{n \choose 4}[/tex] where n s the total number of employees
Step-by-step explanation:
Since [tex]{n \choose x}[/tex] gives the number of ways you can pick a subset of x elements from n.
remember in order to calculate the combination number ( for example if you know the number of office employees) is
[tex]{n \choose 4}=\frac{n!}{4! (n-4)!}[/tex]
using that 4! =1*2*3*4 and same idea for n!
An independent-measures research study uses a total of 18 participants to compare two treatment conditions. If the results are used to construct a 90% confidence interval for the population mean difference, then the t values will be ±1.746.a) Trueb) False
Answers
Answer:
The answer is false
Step-by-step explanation:
In a sample above 30 obs like this the confidence interval is defined as
X+- t* (s/sqrt(n)) where X is the mean t the tvalue for a given confidence level, n the size of sample and s standar deviation.
To find de appropiate value of t we must see the T table where rows are degrees of freedom and columns significance level
The significance is obtained:
significance = 1 - confidence level = 1 - 0.9 = 0.10
Degrees of freedom (df) for the inteval are
df = n - 1 = 18 - 1 = 17
So we must look for the value of a t with 17 values and significance of 0.10 which in t table is 1.740 not 1.746 ( thats the t for 16 df)
The statement is false because the critical t value for a 90% confidence interval depends on the degrees of freedom, which, for an independent measures study with 18 participants split into two groups, would be 16 (18 participants - 2 groups),
The statement regarding an independent-measures research study comparing two treatment conditions with 18 participants constructing a 90% confidence interval and having t values of ±1.746 is assessed as False. The critical t value is determined by the degrees of freedom, which in an independent-measures t-test, is generally calculated as the total number of participants minus the number of groups. With 18 participants divided into two groups, the degrees of freedom would be 16 (18 - 2), assuming equal group sizes. The exact t value for a 90% confidence interval would need to be looked up in a t distribution table or calculated using statistical software, and it would likely differ from ±1.746 for 16 degrees of freedom.
To accurately determine the critical t value, one would consult a t distribution table or statistical software tailored to the specific degrees of freedom for the study. It's important to note that confidence intervals, t-tests, and their interpretations are crucial in research for estimating the range within which the true population parameter lies and for assessing the significance of the findings, respectively.
Help! Simplify (see photo) pls explain
If you don’t know pls don’t answer thanks
Answers
Answer:
[tex]-8 \sqrt{6}[/tex]
Step-by-step explanation:
[tex]4i\sqrt{-24} \\4i\sqrt{-1} *\sqrt{24} \\4i*i*\sqrt{4}*\sqrt{6}\\-4 * 2*\sqrt{6} \\-8 \sqrt{6}[/tex]
.
[tex]\sqrt{-1} =i \\i*i = -1[/tex]
What is the end behavior of the polynomial function?
Answers
Answer:
see below
Step-by-step explanation:
Ordinarily a graph like this will have arrowheads on the ends of the curve, indicating the curve continues on in the same direction. Here, we have to assume the existence of such arrowheads in order to choose a sensible answer.
As x goes to -∞, the curve continues to increase toward +∞. This is not an answer choice.
As x goes to +∞, the curve continues to decrease toward -∞. This matches the third answer choice.
Answer:
As [tex]x[/tex] tend to ∞, tehn [tex]y[/tex] tend to -∞.Step-by-step explanation:
In the graph you can observe the beahaviour of each variable.
Observe, as [tex]x[/tex] tend to ∞, then [tex]y[/tex] tend to -∞. This beahaviour is located in the IV quadrant.
So, in the options, the best answer is "as [tex]x[/tex] tend to ∞, tehn [tex]y[/tex] tend to -∞", because is true.
Other options are just false, for example, as [tex]x[/tex] tend to -∞, [tex]y[/tex] doesn't tendo to -∞.
As [tex]x[/tex] tend to -∞ [tex]y[/tex] doesn't tendo to 0.
Therefore, the right answer is the third choice.
Item 5: Suppose an American worker can make 20 pairs of shoes or grow 100 apples per day. On the other hand, a Canadian worker can produce 10 pairs of shoes or grow 20 apples per day. The opportunity cost for Canada is ___.
Answers
Answer:
The opportunity cost for Canada will be comparatively high in the production of shoes.
Step-by-step explanation:
An American worker can make 20 pairs of shoes or grow 100 apples per day.
A Canadian worker can produce 10 pairs of shoes or grow 20 apples per day.
In easy words, opportunity cost is defined as the value of ones next best alternative.
The opportunity cost for Canada will be comparatively high in the production of shoes.
Matching: write in the correct letter.
line segment AB
Ray AB
The length of line segment AB
Line AB
answer options
a) AB
b) AB with a line above it.
c) AB with a line above it going only left.
d) AB with a line with both arrows on either side.
Answers
ray: c
length: a
line: d
Line segment AB - b) AB with a line above it.
Ray AB - c) AB with a line above it going only left.
The length of line segment AB - a) AB
Line AB - d) AB with a line with both arrows on either side.
To match the given terms with the correct letter options, let's define each term and find its corresponding notation:
Line segment AB: A line segment is a part of a line that is bounded by two distinct end points.Ray AB: A ray starts at one point and extends infinitely in one direction. The length of line segment AB: This refers to the numerical length or distance between points A and B. Line AB: A line extends infinitely in both directions.These notations are crucial in geometry to clearly communicate different types of lines and measurements.
El costo por kilo de queso chihuahua, es de 78. el total de queso comprado el dia anterior fue de 195. que fraccion del total de queso chihuahua queda
Answers
Answer: [tex]\frac{3}{8}\ kg[/tex]
Step-by-step explanation:
The table attached is part of the exercise (Without the table the question was incomplete).
We know that the cost of 1 kilogram of Chihuahua cheese is 78 and the Chihuhua cheese bought the day before cost 195.
The first step is to calculate the amount of cheese bought with 195:
[tex]\frac{195}{78}=\frac{5}{2}\ kg[/tex]
Now, we need to add the fractions provided in the table (This table shows the amount of cheese that was used in that day). Then:
[tex]\frac{1}{2}\ kg+\frac{7}{8}\ kg+\frac{3}{4}\ kg=\frac{17}{8}\ kg[/tex]
Finally, in order to find what fraction of Chihuahua cheese is left, we must subtract [tex]\frac{5}{2}\ kg[/tex] and [tex]\frac{17}{8}\ kg[/tex]:
[tex]\frac{5}{2}\ kg-\frac{17}{8}\ kg=\frac{3}{8}\ kg[/tex]
Two angles are complementary. One angle measures 20 degrees more than the other angle. Find the measure of the LARGER angle. Just type in the answer. Do not type in a variable or the degree symbol.
Answers
Two complementary angles equal 90 degrees.
If one is 20, the larger one would be 90 - 20 = 70 degrees.
Dylan has a good credit score and is planning to apply for a loan. What could negatively affect Dylan’s credit score?
A.
missing a loan payment
B.
not making a down payment
C.
providing collateral
D.
using a cosigner
Answers
Answer:
the right answer is a)missing a loan payment
Step-by-step explanation:
because if you are missing a loan payment, you would have a negative report at the risk centers
Answer:
A. missing a loan payment
Step-by-step explanation:
Whenever someone wish to apply a loan, lenders would consider his/her credit scores when analyzing the application. A good credit score would increase his/her chance to be qualified for the loan. The higher the score qualifies you for a fair interest rates, and also it would reduce the perceived risk.
Considering the question, missing a loan payment would definitely affect his credit score negatively. It would affect the interest rate, loan terms and credit limit.
help asap ( will give brainliest )
Answers
Answer:
B
Step-by-step explanation:
the open circle at -9 indicates that x is "greater than" -9, whereas the filled circle at -5 indicates that x is " less than or equal" to -5
Given that Ray B A bisects ∠DBC, which statement must be true? m∠ABD = m∠ABC AB ≅ BC B is the midpoint of DC. m∠DBC = 90°
Answers
Answer:
A.[tex]m\angle ABD=m\angle ABC[/tex]
Step-by-step explanation:
We are given that a ray BA bisects angle DBC.
We have to find true statement .
Angle bisector property:When a ray bisect any angle then the angles made by bisection of angle are equal.
When ray BA bisects angle DBC
Then, [tex]m\angle ABD=m\angle ABC[/tex]
By angle bisector property.
Therefore, option A is true.
Answer:A.[tex]m\angle ABD=m\angle ABC[/tex]
Answer:
A
Step-by-step explanation:
Because I said this was the answer. I know all.
In the past year, Ann watched 27 movies that she thought were very good. She watched 90 movies over the whole year. Of the movies she watched, what percentage did she think were very good?
Answers
Answer:
30%
Step-by-step explanation:
27 over 90 is equal to 0.3 in decimal form, turn that into percentage by multiplying by 100, and you get 30%
For what values of b are the vectors \langle -46, b, 10 \rangle and \langle b, b^2, b \rangle orthogonal
Answers
Answer:
b = 6 or b = -6 (non-zero vectors)
b = 0 (zero vector)
Step-by-step explanation:
Two vectors [tex]\vec{a}=\langle a_1,a_2,a_3\rangle[/tex] and [tex]\vec{b}=\langle b_1,b_2,b_3\rangle[/tex] are orthogonal if their dot product is equal to 0, or in other words
[tex]a_1\cdot b_1+a_2\cdot b_2+a_3\cdot b_3=0[/tex]
In your case,
[tex]\vec{a}=\langle -46, b, 10\rangle\\ \\\vec{b}=\langle b,b^2,b\rangle[/tex]
Hence, if vectors a and b are orthogonal, then
[tex]-46\cdot b+b\cdot b^2+10\cdot b=0\\ \\-46b+b^3+10b=0\\ \\b^3-36b=0\\ \\b(b^2-36)=0\\ \\b(b-6)(b+6)=0\\ \\b=0\text{ or }b=6\text{ or }b=-6[/tex]
Note, then if b = 0, then [tex]\vec{b}=\langle 0,0,0\rangle[/tex] and zero-vector is orthogonal to any other vectors.
Thus, b = 6 or b = -6.
If a ball is thrown into the air with a velocity of 46 ft/s, its height in feet t seconds later is given by y = 46t − 16t2.
Answers
Answer:
Step-by-step explanation:
you want maximum height reached?
[tex]\frac{dy}{dt} =46-32t\\\\at max. height velocity=0\\ 0=46-32t\\32 t=46\\t=46/32=23/16\\y=t(46-16t)\\ at t=\frac{23}{16} \\ y=\frac{23}{16}(46-16*\frac{23}{16} )\\\\y=\frac{529}{16} ft[/tex]
i. Average velocity for a time period of 0.5 seconds: -26 ft/s. ii. Average velocity for a time period of 0.1 seconds: -36.4 ft/s iii. Average velocity for a time period of 0.05 seconds: 3.2 ft/s. iv. Average velocity for a time period of 0.01 seconds: -18.36 ft/s
To find the average velocity for a given time period, we need to calculate the change in height and divide it by the change in time.
Given the height equation:[tex]y = 46t - 16t^2[/tex]
i. Time period of 0.5 seconds:
Initial time, [tex]t_1 = 2[/tex]
Final time, [tex]t_2 = 2 + 0.5 = 2.5[/tex]
Change in time: [tex]\delta t = t2 - t1 = 2.5 - 2 = 0.5[/tex] seconds
To find the change in height, we substitute the initial and final times into the height equation:
Initial height, [tex]y_1 = 46t_1 - 16t_1^2 = 46(2) - 16(2)^2 = 92 - 64 = 28[/tex] feet
Final height, [tex]y_2 = 46t_2 - 16t_2^2 = 46(2.5) - 16(2.5)^2 = 115 - 100 = 15[/tex]feet
Change in height: [tex]\delta y = y_2 - y_1 = 15 - 28 = -13[/tex] feet
Average velocity: V_avg = Δy / Δt = -13 / 0.5 = -26 ft/s (negative since the ball is moving downward)
ii. Time period of 0.1 seconds:
Initial time,[tex]t_1 = 2[/tex]
Final time, [tex]t_2 = 2 + 0.1 = 2.1[/tex]
Change in time: [tex]\delta t = t_2 - t_1 = 2.1 - 2 = 0.1[/tex]seconds
Initial height, [tex]y_1 = 46t_1 - 16t_1^2 = 46(2) - 16(2)^2 = 92 - 64 = 28[/tex] feet
Final height, [tex]y_2 = 46t_2 - 16t_2^2 = 46(2.1) - 16(2.1)^2 = 96.6 - 72.24 = 24.36[/tex] feet
Change in height: [tex]\delta y = y_2 - y_1 = 24.36 - 28 = -3.64[/tex]feet
Average velocity: [tex]V_{avg} = \delta y / \delta t = -3.64 / 0.1 = -36.4[/tex] ft/s (negative since the ball is moving downward)
iii. Time period of 0.05 seconds:
Initial time, [tex]t_1 = 2[/tex]
Final time, [tex]t_2 = 2 + 0.05 = 2.05[/tex]
Change in time: [tex]\delta t = t_2 - t_1 = 2.05 - 2 = 0.05[/tex] seconds
Initial height, [tex]y_1 = 46t_1 - 16t_1^2 = 46(2) - 16(2)^2 = 92 - 64 = 28[/tex] feet
Final height, [tex]y_2 = 46t_2 - 16t_2^2 = 46(2.05) - 16(2.05)^2 =95.4 - 67.24 = 28.16[/tex] feet
Change in height: [tex]\delta y = y_2 - y_1 = 28.16 - 28 = 0.16[/tex] feet
Average velocity: [tex]V_{avg} = \delta y / \delta t = 0.16 / 0.05 = 3.2[/tex] ft/s
iv. Time period of 0.01 second:
Initial time, t_1 = 2[tex]t_1 = 2[/tex]
Final time, [tex]t_2 = 2 + 0.01 = 2.01[/tex]
Change in time: [tex]\delta t = t2 - t1 = 2.01 - 2 = 0.01[/tex]seconds
Initial height, [tex]y_1 = 46t_1 - 16{t_1}^2 = 46(2) - 16(2)^2 = 92 - 64 = 28[/tex] feet
Final height, [tex]y_2 = 46t_2 - 16t2^2 = 46(2.01) - 16(2.01)^2 =92.46 - 64.6436 = 27.8164[/tex] feet
Change in height: [tex]\delta y = y_2 - y_1 = 27.8164 - 28 = -0.1836[/tex] feet
Average velocity: [tex]V_avg = \delta y / \delta t = -0.1836 / 0.01 = -18.36[/tex] ft/s (negative since the ball is moving downward)
To estimate the instantaneous velocity when t = 2, we can find the derivative of the height equation with respect to time, dy/dt:
[tex]y = 46t - 16t^2\\dy/dt = 46 - 32t[/tex]
Substitute t = 2 into the derivative equation:
[tex]dy/dt = 46 - 32(2) = 46 - 64 = -18[/tex] ft/s (negative since the ball is moving downward)
Therefore, the estimated instantaneous velocity when t = 2 is -18 ft/s.
Hence, i. Average velocity for a time period of 0.5 seconds: -26 ft/s. ii. Average velocity for a time period of 0.1 seconds: -36.4 ft/s iii. Average velocity for a time period of 0.05 seconds: 3.2 ft/s. iv. Average velocity for a time period of 0.01 seconds: -18.36 ft/s
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Which shows the phrase "the difference between a number and 10" as a variable expression?
x−10
10x−10
x + 10
−10x
Answers
Answer: x-10
Since you're finding the difference of a number, x, and 10, you would do x-10.
Final answer:
The variable expression for "the difference between a number and 10" is represented by x − 10, which implies the subtraction of 10 from a variable x.
Explanation:
The phrase "the difference between a number and 10" as a variable expression is represented by x − 10. When we discuss 'the difference' in mathematical terms, we are referring to the result of subtracting one number from another. Here, whatever the unknown number (represented by the variable x) is, we are subtracting 10 from it.
For which values of λ does the system of equations (λ − 2)x + y = 0 x + (λ − 2)y = 0 have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.
Answers
Answer:
λ=3,λ=1
Step-by-step explanation:
let (λ-2)=a
[tex]ax + y = 0\\x + ay = 0[/tex]
solve:
[tex]ax + y = 0\\ax + a^2y = 0\\y-a^2y=0\\a^2 = 1[/tex]
replace a:
[tex](\lambda-2)^2=1\\\lambda^2-4\lambda+4=1\\\lambda^2-4\lambda+3=0[/tex]
solve:
[tex]\lambda_1=2+\sqrt{4-3} =3\\\lambda_2=2-\sqrt{4-3}=1[/tex]
Nontrivial solutions in a system of equations are found when the determinant of the characteristic matrix is zero. For the given equations, the values of λ that lead to nontrivial solutions are λ = 2, with a nontrivial solution x = 1, y = -1.
Nontrivial solutions of the system of equations occur when the determinant of the characteristic matrix is zero. For the given equations, you need to find values of λ that make the determinant of the matrix zero. This means solving for λ where (λ-2)^2 = 0.
Thus, the values of λ that lead to nontrivial solutions are λ = 2. For λ = 2, a nontrivial solution would be x = 1, y = -1.
This process identifies when the system has solutions beyond the trivial one where x = y = 0.
A ball is thrown in the air from a ledge. It's height in feet represented by f(x)=16(x^2-6x-7), where x is the number of seconds since the ball has been thrown. The height of the ball is 0 feet when it hits the ground. How many seconds does it take the ball to reach the ground?
Answers
Answer:
Step-by-step explanation:
Since we know that the height is 0, we can figure out how long it took the ball to reach the ground by setting [tex]f(x) = 0[/tex] and solving for [tex]x[/tex]:
[tex]f(x) = 16(x^{2} - 6x - 7)[/tex]
[tex]0 = 16(x^{2} - 6x - 7)[/tex]
[tex]0 = x^{2} - 6x - 7[/tex]
[tex]0 = (x - 7)(x + 1)[/tex]
[tex]x = -1, 7[/tex]
Because time can only be positive, the answer is 7 seconds.
Answer:
7 seconds.
Step-by-step explanation:
height h = 16(x^2-6x-7) = 0
x^2 - 6x - 7 = 0
(x - 7)(x + 1) = 0
x = 7 seconds (we ignore the negative).
In a housing project there are 350 households in which English is spoken, 50 in which Spanish is spoken, and 100 in which the language is other than English or Spanish. If a psychologist approaches a house at random to conduct an interview, the chance that the language in that household will NOT be English?a- .002b- .14c- .3d- .43
Answers
Answer: C. 0.3
Step-by-step explanation:
Given : Number of English speaking households =350
Number of Spanish speaking households =50 (1)
Number of households in which the language is other than English or Spanish=100 (2)
Total households participates in this survey =350+50+100=500
No. of households not speak English =100+50=150 (Add (1) and (2))
If a psychologist approaches a house at random to conduct an interview, the chance that the language in that household will NOT be English will be :_
[tex]\dfrac{\text{No. of households not speak English}}{\text{Total households}}\\\\=\dfrac{150}{500}\\\\=\dfrac{3}{10}=0.3[/tex]
Hence, the correct option is option (c).
For your rock collection display you want to have at most 25 samples. You want to have at least 3 times as many sedimentary samples (x) as metamorphic samples (y)
Answers
Answer:
Step-by-step explanation:
A certain one-day seminar consisted of a morning session and an afternoon session. If each of the 128 people attending the seminar attended at least one of the two sessions, how many of the people attended the morning session only?
Answers
Answer: 64 people attended to the morning session only.
Step-by-step explanation:
They told us that each one of the 128 people attended at least one of the two sessions of the one-day seminar. We don't know for sure to which one of the sessions they attended, we only know that every person attended at least one. The probability of one person going to the morning session is the same as the probability that they will go to the afternoon session: 50%. To get the number of persons that attended the morning session only, we simply have to perform the product between the probability and the total number of potential attendees to the seminar. Let N be the total number of attendes, M the number of persons going to the morning session only and P the probability of those persons actually going to that session:
[tex]M = N \times P = 128 \times 0.5 = 64[/tex]
So the total number of persons that attended the morning sessions only is 64.
Let 5 be the region that lies between the curves y=− xm ; y= − xn ; 0 < x < 1 where m and n are integers with 0 < n , m. (a) Sketch the region 5. (b) Find the coordinates of the centroid of 5. (c) Try to find values of m and n such that the centroid lie.
Answers
Answer:
(a) Please see the first figure attached
(b) The coordinates of the centroid are [tex]G(\frac{2}{3}, \frac{m+n}{3} )[/tex]
(c) due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of [tex]m[/tex] and [tex]n[/tex], the centroid will lie
Step-by-step explanation:
Hi, let us first solve part (a). Since for any given values of [tex]n[/tex] and [tex]m[/tex] we will obtain two linear functions:
[tex]y=-mx[/tex] and
[tex]y=-nx[/tex]
with [tex]0\leq x\leq 1[/tex] we can assure that our region is going to be a triangle. To see this, please take a look at the plot I generated using Wolfram. In this case, I have used two specific values for m and n but keeping the condition [tex]0\leq n\leq m[/tex].
Now, for part (b) let me start remembering what the centroid is: the centroid of a triangle is the point where the three medians of the triangle meet. And a median of a triangle is a line segment from one vertex to the midpoint on the opposite side of the triangle (see the second figure where the medians are depicted in red and the centroid of the triangle, G is depicted in blue). For a given triangle [tex]\bigtriangleup \rm{ABC}[/tex], the coordinates of its centroid [tex]G[/tex] are given by:
[tex]G_x=\frac{A_x+B_x+C_x}{3}[/tex] and [tex]G_y=\frac{A_y+B_y+C_y}{3}[/tex]
Now let's apply this to our problem. Take a look at the first figure. The vertex A has clearly coordinates [tex](0,0)[/tex] for any value of [tex]m[/tex] and [tex]n[/tex] since the two lines have their intersection with y-axis in this point.
To obtain the coordinates of [tex]B[/tex] and [tex]C[/tex], let's use the given functions and the fact that the coordinate x is limited to 1. Then, we have:
For A:
[tex]y=-mx[/tex] then, when [tex]x=1[/tex], substituting in the formula [tex]y=-m[/tex]
For B and doing the same as for A:
[tex]y=-nx[/tex] then, when [tex]x=1[/tex], substituting in the formula [tex]y=-n[/tex]
Thus, the coordinates of the vertices of the triangle are: [tex]A(1, m)[/tex], [tex]B(1,3)[/tex] and [tex]C(0,0)[/tex] and the coordinates of the centroid are:
[tex]G_x=\frac{A_x+B_x+C_x}{3} = G_x=\frac{1+1+0}{3}\\G_x=\frac{2}{3}[/tex]
and
[tex]G_y=\frac{A_y+B_y+C_y}{3}=\frac{m+n+0}{3}\\G_y=\frac{m+n}{3}[/tex].
Summarizing: the coordinates of the centroid of the region are [tex]G(\frac{2}{3}, \frac{m+n}{3} )[/tex]
Now, for part (c), due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of [tex]m[/tex] and [tex]n[/tex], the centroid will lie. Other important points of the triangle, like the orthocentre and circumcentre, can lie outside in obtuse triangles. In right triangles, the orthocentre always lies at the right-angled vertex.