The lengths of the legs of a similar triangle to the one Juan drew, which has leg lengths 6cm and 8cm, could be 12 cm and 16 cm because the sides of similar triangles are proportional.
Explanation:Juan's right triangle has leg lengths of 6 centimeters and 8 centimeters. To find a similar right triangle, we need ratios of corresponding sides to be equivalent or proportional. As these are right triangles, we can use the Pythagorean theorem which states that a² + b² = c². If the original triangle has sides 6 cm and 8 cm, then triangle with proportional sides could have sides that are multiple of these dimensions, such as 12 cm (which is 6cm x 2) and 16 cm (which is 8cm x 2). So, one example of a similar triangle could have leg lengths of 12 cm and 16 cm.
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A 180-watt iHome® is used on an average of three hours a day. Find the cost of listening to the iHome for one week at a cost of $0.13 per kilowatt-hour.
A. $0.49
B. $11.34
C. $491.40
D. $0.07
The answer is:
The correct option is:
A. $0.49
Why?From the statement, we know that the iHome is used on average three hours a day, and we are asked to find the cost for a week, so first, we need to calculate the total hours that the iHome is used for, and then, calculate the kilowatt-hour consumption rate.
[tex]TotalTime_{week}=3\frac{hours}{day} *7days=21hours[/tex]
[tex]TotalEnergyConsumption_{week}=180watt*21hours=3780watt.hour[/tex]
Now, we must remember that:
[tex]1Kilowatt=1000watts[/tex]
So,
[tex]3780watts=3780watts.hour*\frac{1KiloWatt}{1000watts}=3.78KiloWatt.hour[/tex]
Then, calculating the cost, we have:
[tex]TotalCost_{week}=0.13\frac{dollar}{killowat.hour}*3.78killowat.hour=0.49(dollar)[/tex]
Hence, we have that the correct option is:
A. $0.49
Have a nice day!
3. A projectile is fired from ground level with an initial velocity of 35 m/s at an angle of 35° with the horizontal. How long will it take for the projectile to reach the ground?
Answer: 4.10s
t=2Vi*sin(theta)/g
Vi=initial velocity=35m/s
g=9.8m/s^2
t=2*35*sin(35)/9.8=4.10s
Answer:
4.1 seconds to the nearest tenth.
Step-by-step explanation:
The vertical component of the velocity = 35 sin 4.935 m/s.
The relation between the height (h) of the projectile and time is given by:
h = ut + 1/2 g^2 where u = initial velocity, t = the time and g = acceleration due to gravity which we can take to be 9.8 m/s/s. When the projectile hits the ground h = 0 .
So we have h = 35sin35 t - 4.9t^2 = 0
t(35sin35 - 4.9t) = 0
4.9t = 35 sin35
t = 35 sin 35 / 4.9
= 4.097 seconds
Sometimes you to ___ some points to get a good approximation of the location of extreme values
Answer:
Sometimes you to plot some points to get a good approximation of the location of extreme values.
Approximating some points, especially influential points or outliers, helps to get a better grasp of the location of the extreme values. This is crucial in fields like calculus, statistics and graphing, and it assists in identifying patterns and trends. Additionally, substitution of 'x' values in an equation can help estimate 'y' values.
Explanation:Sometimes in mathematics, particularly in fields such as calculus, statistics, and graph theory, you need to approximate some points for a good understanding of the location of extreme values. This practice is especially useful when identifying influential points or outliers within a data set, as these points significantly alter the slope or fitness of a regression line. When you find such points, you can exclude them from your calculations to get a more accurate overview of the general pattern or trend.
In the case of graphing, selection of an appropriate scale for both axes is also crucial. The scale should reflect all your data while also making it easy to identify any trends within it. Too large a scale can make data changes hard to see, while a too fine scale necessitates more space for the graph and can crowd information.
Additionally, in calculus and algebra, to estimate the 'y' values for various 'x-values', one can substitute the 'x' values into the equation. This helps to offer a clearer insight into the correlation of variables within a function.
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What are the answers to these? List them by order from top to bottom please, also the top question is that they want the design that will use less cardboard to make, which box is it? ❤️❤️❤️
Answer:
proposed design uses less cardboard, also has less volume. I don't dig it.BOXer B has the greatest surface areaBOXer D has the greatest surface areaStep-by-step explanation:
When there are many instances of the same calculation, it is convenient to let a spreadsheet or graphing calculator do them. The formula can be entered once and used many times. See the attachment for an example.
1. The surface area of a box with dimensions L, W, D can be written as ...
S = 2(LW +LD +WD) = 2(LW +D(L+W))
Then the surface area of the left (original) box is ...
S = 2(2·12 + 8(2+12)) = 2(24 +112) = 272 . . . . square inches
The surface area of the right (proposed) box is ...
S = 2(4·3 +14(4+3)) = 2(12 +98) = 220 . . . . square inches
The volume of the original box, at 2·12·8 = 192 in³ is greater than the volume of the proposed box (3·4·14 = 168 in³), so the customer gets less cereal with the redesigned box. I don't dig it.
__
2. The previous question shows the formula and an example of the calculation. The attachment shows the numbers for this question.
box A: 62 ft²box B: 70 ft² — winner__
3. The attachment shows the numbers for this question.
box C: 270 m²box D: 272 m² — winnerJivesh also has a more powerful Model B rocket. For this rocket, he uses the equation h=-490t^2+1260t. When is the height of the Model B rocket 810 centimeters? ( it also includes number 19, but I need 20)
Answer:
1.29 s
Step-by-step explanation:
h = -490t² + 1260t
810 = -490t² + 1260t
490t² - 1260t + 810 = 0
49t² - 126t + 81 = 0
(7t - 9)² = 0
7t - 9 = 0
t = 9/7
t ≈ 1.29
To find the time when the rocket is at a height of 810 cm, set the equation -490t^2 + 1260t equal to 810 and solve for t using the quadratic formula. The quadratic formula will give two solutions: choose the positive solution as we cannot have negative time.
Explanation:We are given the equation h=-490t^2+1260t to represent the height of the rocket. We're also told that the height h is 810 cm and we are asked to solve for time t when this is the case.
To find the time when the rocket's height is 810 cm, we can first set the equation equal to 810: -490t^2 + 1260t = 810.
We can then simplify that to -490t^2 + 1260t - 810 = 0 and solve for t using the quadratic formula t = [-b ± sqrt(b² - 4ac)] / 2a, where a = -490, b = 1260 and c = -810.
This will give us the two points in time at which the rocket is at a height of 810 cm. Remember, the negative solution is likely extraneous as we cannot have negative time, so you should consider only the positive solution.
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The function below shows the number of car owners f(t), in thousands, in a city in different years t:f(t) = 1.1t2 − 2.5t + 1.5The average rate of change of f(t) from t = 3 to t = 5 is ______ thousand owners per year.Answer for Blank 1:
Answer:
The average rate of change is : [tex]6.3[/tex]
Step-by-step explanation:
The number of car owners is modeled by the function;
[tex]f(t)=1.1t^2-2.5t+1.5[/tex], where t is the different number of years.
The average rate of change of f(t) from t=3 to t=5 is simply the slope of the secant line connecting:
(3, f(3)) and (5,f(5))
Which is given by:
[tex]\frac{f(5)-f(3)}{5-3}[/tex]
Now, we substitute t=3 into the function to get;
[tex]f(3)=1.1(3)^2-2.5(3)+1.5[/tex]
[tex]f(3)=3.9[/tex]
We substitute t=5 into the function to get;
[tex]f(5)=1.1(5)^2-2.5(5)+1.5[/tex]
[tex]f(5)=16.5[/tex]
Therefore the average rate of change is : [tex]\frac{14.5-3.9}{2}=6.3[/tex]
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
Tammy knows that the factors of a polynomial are ƒ(x) = (x − 2)(x − 1)(x + 4) and the solutions are x = −4, 1, and 2. Which graph models this polynomial?
The answer would be D, where all of the given x values touch the x axis
Answer: D)
Step-by-step explanation:
The equation is: (x - 2)(x - 1)(x + 4) = 0
We can find the x-intercepts by Applying the Zero Product Property:
x - 2 = 0 x - 1 = 0 x + 4 = 0
x = 2 x = 1 x = -4
Which graph shows the curve crossing the x-axis at these points? D
In a certain region, the equation y=19.485x+86.912 models the amount of a homeowner’s water bill, in dollars, where x is the number of residents in the home.
What does the slope of the equation represent in context of the situation?
1. The water bill increases by about $19 every month.
2. The water bill increases by about $19 for every additional resident in the home.
3. The water bill increases by about $87 every month.
4. The water bill increases by about $87 for every additional resident in the home.
Answer:
It's choice 2.
Step-by-step explanation:
y=19.485x+86.912
The 19.485 is the slope of the graph of this equation. This gives the rate of change of the amount of the bill (above $86.912) for each added resident (x).
Help with Algebra! (Photo attached)
Answer:
D. The graph of g(x) is shifted 2 units up.
Step-by-step explanation:
Adding 2 to the y-coordinate of a point shifts it up by 2 units.
___
The graph of f(x) is all points (x, f(x)). When you add 2 to f(x), you make the graph of g(x) be all points (x, g(x)) = (x, f(x)+2). That is all of the points on the original graph are shifted up by 2 units.
How do you use a system of equations to find the solution algebraically?
Answer:
Pemdas
Step-by-step explanation:
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
You go from left to right and solve in the order called Pemdas.
To use a system of equations to find the solution algebraically, follow these steps: identify the unknowns and knowns, write down the equations, solve for one variable, substitute the expression into the other equation(s), solve for the remaining variables, and check your answer(s) for reasonableness.
Explanation:Using a System of Equations to Find the Solution Algebraically1. Identify the unknowns and knowns.
2. Write down the equations that represent the given information.
3. Solve one of the equations for one variable in terms of the other.
4. Substitute this expression into the other equation(s), replacing the variable.
5. Solve the resulting equation(s) to find the value(s) of the remaining variable(s).
6. Check your answer(s) to ensure they make sense in the context of the problem.
In a fruit cocktail, for every 25 ml of orange juice you need 30 ml of apple juice and 45 ml of coconut milk. What proportion of the cocktail is coconut milk? Give your answer as a fraction in its simplest form.
Answer:
9 / 20
Step-by-step explanation:
We can form a ratio with the information "for every 25 ml of orange juice you need 30 ml of apple juice and 45 ml of coconut milk"
25 : 30 : 45
Total = 100
Coconut milk = 45 ml
45 / 100 = 9 / 20
One jar of jelly costs $2.32 for 16 ounces. Another jar costs $2.03 for 13 ounces. Which is the better buy? Why? The jelly that costs $____ for ____ ounces is the better buy. The unit rate for this jar of jelly is $____, or approximately $____ per ounce. The unit rate for the second jar of jelly is $____, or approximately $____ per ounce. Question 4 options: Blank # 1 Blank # 2 Blank # 3 Blank # 4 Blank # 5 Blank # 6
Answer:
The jar of jelly that costs $2.32 for 16 ounces is the better buy, because the unit rate is less
Step-by-step explanation:
step 1
Find the units rate
One jar of jelly costs $2.32 for 16 ounces
so
The unit rate is equal to [tex]\frac{2.32}{16}= 0.145\frac{\$}{ounce}[/tex]
Another jar costs $2.03 for 13 ounces
so
The unit rate is equal to [tex]\frac{2.03}{13}= 0.156\frac{\$}{ounce}[/tex]
step 2
Compare the unit rates
[tex]0.145\frac{\$}{ounce} < 0.156\frac{\$}{ounce}[/tex]
therefore
The jar of jelly that costs $2.32 for 16 ounces is the better buy, because the unit rate is less
The jelly that costs $2.32 for 16 ounces is the better buy. The unit rate for this jar of jelly is $0.145 or approximately $0.15 per ounce. The unit rate for the second jar of jelly is $0.156 or approximately $0.16 per ounce
Tickets cost $4.75 for adults and $2.50 for children What is the total cost of the tickets for two adults and three children
Answer:
$9.50 - for adults
$7.50 - children
Step-by-step explanation:
All you have to do is multiply $4.75 by 2 since there are two adults and multiply $2.50 by 3 since there are three children.
Please please help!!
Answer:
240.3
Step-by-step explanation:
Tan(31) = y/400ft
400 tan (31) = y
y = 240,3
Line l is parallel to line m. The slope of? line l is 8/5 . What is the slope of line m?
Answer:
8/5
Step-by-step explanation:
The slope of two parallel lines is always the same, just are in different locations on the x or y axis.
Seth and Eva are biking on a trail. Seth begins 8 miles ahead of Eva and bikes at an average speed of 4 miles per hour. Eva bikes at an average speed of 6 miles per hour. How much time will it take for Eva to catch up with Seth on the trail?
Answer:
4 hours
Step-by-step explanation:
0:8
6:12
12:16
18:20
24:24
Answer: 4 hours
Step-by-step explanation:
"No solutions & all real numbers"
Solve each equation showing all work:
1.) -2(6 - 2x) = 4(-3 + x)
2.) 5 - 1(2x + 3) = -2(4 + x)
Answer:
1.)[tex]-12+4x=-12+4x[/tex]
2.) [tex]2-2x \neq -8-2x[/tex]
Step-by-step explanation:
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding all the products.
[tex]a(b+c)=ab+ac[/tex]
For the first equation
[tex]-2(6-2x)=4(-3+x)\\(-2)(6)+(-2)(-2x)=(4)(-3)+(4)(x)\\-12+4x=-12+4x[/tex]
For the second equation
[tex]5-1(2x+3)=-2(4+x)\\5+((-1)(2x)+(-1)(3))= (-2)(4)+(-2)(x)\\5+(-2x-3) = -8+(-2x)\\5-2x-3=-8-2x\\2-2x \neq -8-2x[/tex]
.
Bob has 35 liters of lemonade if he distributes all the lemonade equally into 7 juice pitchers, how much lemonade will be in each pitcher?
Answer:
5 liters
Step-by-step explanation:
Divide 35 liters evenly between the 7 pitchers and you'll have 5 liters in each pitcher.
To find the amount of lemonade in each pitcher, divide the total amount of lemonade by the number of pitchers. In this case, each pitcher will contain 5 liters of lemonade.
Explanation:To find the amount of lemonade in each pitcher, we divide the total amount of lemonade by the number of pitchers. In this case, Bob has 35 liters of lemonade and 7 pitchers. So, to find the amount of lemonade in each pitcher, we divide 35 by 7.
35 ÷ 7 = 5 liters of lemonade
Therefore, there will be 5 liters of lemonade in each pitcher.
If Bob has 35 liters of lemonade and he distributes it equally into 7 juice pitchers, we need to perform a division to find out how much lemonade will be in each pitcher. The calculation is straightforward:
Divide the total volume of lemonade by the number of pitchers.
35 liters ÷ 7 pitchers = 5 liters per pitcher.
Therefore, each juice pitcher will contain 5 liters of lemonade.
The table shows the estimated number of deer living in a forest over a five-year period. Are the data best represented by a linear, exponential, or quadratic model? Write an equation to model the data.
0/89 1/55 2/34 3/21 4/13
a. quadratic; y = 0.62x2 + 89
b. exponential; y = 89 • 0.62x
c. linear; y = 0.62x + 89
d. quadratic; y = 89x2 + 0.62
Answer:
exponential; y = 89 • 0.62^x
Step-by-step explanation:
Answer:
Option B exponential y = 89 · 0.62x
Step-by-step explanation:
The table shows the estimated number of deer living in a forest over a five year period.
Year Number of deers
0 89
1 55
2 34
3 21
4 13
Now we have to find the model representing this situation. Difference in number of deer, in the forest.
We can see there is a common ratio between each successive term r = [tex]\frac{55}{89}[/tex] = 0.618
r = [tex]\frac{34}{55}[/tex] = 0.618
so it can be represented by an exponential model.
[tex]y=a (r) ^{x}[/tex]
[tex]y=89(62) ^{x}[/tex]
Option B is the answer.
I have a vase that is shaped like a rectangular prism. The height is 10 inches. The length of the base is 6 inches. The width of the base is 5 inches. What is 2/3 of the volume of the vase? If necessary, round to the nearest whole number.
Answer:
[tex]200in^3[/tex]
Step-by-step explanation:
To solve this, we are using the formula for the volume of a rectangular prism:
[tex]V=whl[/tex]
where
[tex]w[/tex] is the width of the base
[tex]l[/tex] is the length of the base
[tex]h[/tex] is the height of the prism
We know from our problem that the height is 10 inches. The length of the base is 6 inches. The width of the base is 5 inches.
Replacing values
[tex]V=(5in)(10in)(6in)[/tex]
[tex]V=300in^3[/tex]
Now, to find 2/3 of that volume, we just need to multiply it by 2/3
[tex]\frac{2}{3} V=\frac{2}{3} 300in^3[/tex]
[tex]\frac{2}{3} V=200in^3[/tex]
We can conclude that 2/3 of the volume of the vase is 200 cubic inches.
Please give details on how to do this:
Given the translation T(-2, 5), translate the given ordered pairs: (2, 5) and (-1, 7
Answer:
It is 15.
Step-by-step explanation:
a chef plans to mix 100% vinegar with italian dressing. the italian dressing contains 7% vinegar. the chef wants to make 310 milliliters of a mixture that contains 19% vinegar. how much vinegar and how much italian dreessing should she use ?
vinegar: ? milliliters
italian dressing: ? milliliters
Let [tex]x[/tex] be the amount (mL) of the pure vinegar the chef will use, and [tex]y[/tex] the amount of dressing. She wants to end up with a 310 mL mixture, so
[tex]x+y=310[/tex]
For each mL used of the dressing, 0.07 mL is vinegar, and the chef wants to end up with a 19% vinegar mixture, so
[tex]x+0.07y=0.19(x+y)=58.9[/tex]
Now
[tex]x+y=310\implies y=310-x[/tex]
[tex]\implies x+0.07(310-x)=58.9[/tex]
[tex]\implies0.93x+21.7=58.9[/tex]
[tex]\implies0.93x=37.2[/tex]
[tex]\implies x=40[/tex]
[tex]\implies y=270[/tex]
What is the probability of getting a spade or a red card?
Answer:
3/4
Step-by-step explanation:
1/2 the deck is red cards.
1/4 of the deck is spades.
There are no spades that are red cards, so the probability of drawing a red card or a spade at random from a well-shuffled deck is ...
1/2 + 1/4 = 3/4
Two search teams spot a stranded climber on a mountain. The first search team is 0.5 miles from the second search team. If the angle of elevation from the first search team to the stranded climber is 15° and the angle of elevation from the second search team is 22° to the stranded climber, what is the altitude of the climber if both search teams are standing at an altitude of 1 mile high?
Answer:
The altitude of the climber is 1.40 miles
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
In the right triangle BCD
tan(22°)=h/(x-0.5)
h=tan(22°)*(x-0.5) ----> equation A
In the right triangle ACD
tan(15°)=h/x
h=tan(15°)*(x) ----> equation B
Equate equation A and equation B and solve for x
tan(22°)*(x-0.5)=tan(15°)*(x)
tan(22°)*x-tan(22°)*0.5=tan(15°)*x
x[tan(22°)-tan(15°)]=tan(22°)*0.5
x=tan(22°)*0.5/[tan(22°)-tan(15°)]
x=1.48 miles
Find the value oh h
h=tan(15°)*(1.48)=0.40 miles
therefore
The altitude of the climber is equal to
0.40+1=1.40 miles
Answer:
Step-by-step explanation:
Just solved a similar problem and figured it out, the angle of elevation is somewhat unintuitive as the angle of the second search team needs to be flipped. The diagram should look more like this:
This isn't an explanation of the math but more of a visualization for those that just needed the 2D representation. The real answer would be 1.081 miles (:
A set of telephone poles is stacked in a pile, 8 layers high. The top layer consists of 20 telephone
poles. The next layer down consists of 24 telephone poles. The third layer consists of 28
telephone poles. If this pattern continues for the remaining 5 layers, how many telephone poles
are in the pile?
A. 224
B. 244
C. 252
D. 272
Answer:
D. 272 poles
Explanation:
We are given that:
The top layer has 20 poles, the next down one has 24 poles and the third one has 28 poles
We can note that each layer has 4 poles more that the one above it
Based on this, we can get the number of poles in each layer as follows:
Top layer has 20 poles
Second one has 20 + 4 = 24 poles
Third one has 24 + 4 = 28 poles
Fourth one has 28 + 4 = 32 poles
Fifth one has 32 + 4 = 36 poles
Sixth one has 36 + 4 = 40 poles
Seventh one has 40 + 4 = 44 poles
Eighth one has 44 + 4 = 48 poles
Now, we can get the total number of poles by adding the poles in all layers
This is done as follows:
Total number of poles = 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48
Total number of poles = 272 poles
Hope this helps :)
The manager of a frozen yogurt shop wants to add some new flavors that will appeal to customers. Which surveying method is most likely to produce a representative sample of the yogurt shop's customers?
Final answer:
The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is true random sampling.
Explanation:
The surveying method that is most likely to produce a representative sample of the yogurt shop's customers is the true random sampling method. This method involves randomly selecting participants from the entire population of yogurt shop customers, ensuring that each customer has an equal chance of being selected. This helps to minimize bias and ensure that the sample is representative of the entire customer population.
For example, the manager can generate a list of all the customers who have made purchases at the yogurt shop over a specific period of time, and then use a random number generator or a random selection method to choose a certain number of customers from the list. This will ensure that the selected participants represent the diversity of the yogurt shop's customer base.
The most appropriate surveying method for the yogurt shop manager to obtain a representative sample of customers is systematic sampling.
To obtain a representative sample of the yogurt shop's customers, the most appropriate surveying method would be a systematic sampling approach. This involves selecting every nth customer who visits the shop during different times of the day and different days of the week.
By using a systematic sampling method, the manager can ensure that the sample includes customers from various demographic groups, such as different age ranges, genders, and visiting patterns. This approach reduces the potential for bias that may arise from convenience sampling methods, where only customers who are readily available or willing to participate are surveyed.
Additionally, the systematic sampling method allows the manager to capture the preferences of both regular and occasional customers, as well as those who visit during peak and off-peak hours. This comprehensive representation of the customer base increases the likelihood that the survey results will accurately reflect the preferences of the yogurt shop's overall customer population.
Systematic sampling is more time-consuming and resource-intensive than convenience sampling, but it is a more reliable method for obtaining a representative sample of the yogurt shop's customers, which is crucial for making informed decisions about introducing new flavors that will appeal to a wide range of customers.
There are rabbits and chickens in Sally's backyard: 22 heads and 76 legs. How many rabbits and how many chickens are there?
there is 6 chickens and 16 rabbits, need the work tho?
6 chickens 16 rabbits
Match the one-to-one functions with their inverse functions.
I'll match them for you, but to find the inverse of an equation, all you must do is
Switch x and y Solve for y again for the "inverse" ![tex]f(x)^{-1} = 5x[/tex] → [tex]f(x) = \frac{x}{5}[/tex]
[tex]f(x)^{-1} = \frac{x^{3}}{2}[/tex] → [tex]f(x) = \sqrt[3]{2x}[/tex]
[tex]f(x)^{-1} = x + 10[/tex] → [tex]f(x) = x - 10[/tex]
[tex]f(x)^{-1} = \frac{3(x+17)}{2}[/tex] → [tex]f(x) = \frac{2x}{3} -17[/tex]
Hope I help ! :)
ANSWER
[tex] \boxed {f(x)= \frac{2x}{3} - 17\to \: f ^{ - 1} (x)=\frac{3x + 51}{2}}[/tex]
[tex] \boxed {f(x) = x - 10 \to {f}^{ - 1} (x) = x + 10 }[/tex]
[tex] \boxed {f(x) = \sqrt[3]{2x} \to {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} }[/tex]
[tex] \boxed {f(x) = \frac{x}{5} \to{f}^{ - 1} (x) = 5x}[/tex]
EXPLANATION
1.
Given :
[tex]f(x) = \frac{2x}{3} - 17[/tex]
Let
[tex]y =\frac{2x}{3} - 17[/tex]
Interchange x and y.
[tex]x=\frac{2y}{3} - 17[/tex]
Solve for y.
[tex]x + 17=\frac{2y}{3} [/tex]
[tex]3x + 51=2y[/tex]
[tex]y=\frac{3x + 51}{2} [/tex]
[tex]f ^{ - 1} (x)=\frac{3x + 51}{2} [/tex]
2.
Given: f(x)=x-10
Let y=x-10
Interchange x and y.
x=y-10
Solve for y.
y=x+10
This implies that,
[tex] {f}^{ - 1} (x) = x + 10[/tex]
3.
Given:
[tex]f(x) = \sqrt[3]{2x} [/tex]
Let
[tex]y=\sqrt[3]{2x} [/tex]
Interchange x and y.
[tex]x=\sqrt[3]{2y} [/tex]
solve for y.
[tex] {x}^{3} = 2y[/tex]
[tex]y = \frac{ {x}^{3} }{2} [/tex]
[tex] {f}^{ - 1} (x) = \frac{ {x}^{3} }{2} [/tex]
4.
Given:
[tex]f(x) = \frac{x}{5} [/tex]
Let
[tex]y = \frac{x}{5} [/tex]
Interchange x and y.
[tex]x = \frac{y}{5} [/tex]
Solve for y.
[tex]y = 5x[/tex]
[tex] {f}^{ - 1} (x) = 5x[/tex]
A car will be traveling a total distance of 520 miles. The first part of the trip takes 2 hours, and the car’s average rate is 65 miles per hour. If the entire trip takes 8 hours, what is the car’s average rate, in miles per hour, during the second part of the trip?
49
57
50
65
Final answer:
The car's average rate during the second part of the trip is 65 miles per hour, which is calculated by dividing the remaining distance of 390 miles by the remaining travel time of 6 hours.
Explanation:
We need to find the average rate of the car during the second part of the trip. We know the total distance of the trip is 520 miles and the total time for the trip is 8 hours. Since the first part of the trip was at 65 miles per hour for 2 hours, the car would have covered 130 miles (65 miles/hour * 2 hours).
We subtract the distance covered in the first part of the trip from the total distance to find the distance covered during the second part of the trip: 520 miles - 130 miles = 390 miles. The remaining time for the second part of the trip is 8 hours - 2 hours = 6 hours.
To find the average speed during the second part of the trip, we divide the remaining distance by the remaining time:
Average rate = Distance / Time = 390 miles / 6 hours = 65 miles per hour.
Solve the following equation for y.
2y + 2 = 36
Answer:
y = 3 • ± √2 = ± 4.2426
Step-by-step explanation:
2y2 - 36 = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
2y2 - 36 = 2 • (y2 - 18)
Trying to factor as a Difference of Squares :
3.2 Factoring: y2 - 18
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 18 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Equation at the end of step 3 :
2 • (y2 - 18) = 0
Step 4 :
Equations which are never true :
4.1 Solve : 2 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation :
4.2 Solve : y2-18 = 0
Add 18 to both sides of the equation :
y2 = 18
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
y = ± √ 18
Can √ 18 be simplified ?
Yes! The prime factorization of 18 is
2•3•3
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 18 = √ 2•3•3 =
± 3 • √ 2
The equation has two real solutions
These solutions are y = 3 • ± √2 = ± 4.2426