Answer:
A = 68 unit^2
Step-by-step explanation:
Given:-
The piece-wise function f(x) is defined over an interval as follows:
f(x) = { x^2+3x+4 , x < 3
f(x) = { x^2+3x+4 , x≥3
Domain : [ -3 , 4 ]
Find:-
Find the area of the region enclosed by f(x) and the x axis
Solution:-
- The best way to tackle problems relating to piece-wise functions is to solve for each part individually and then combine the results.
- The first portion of function is valid over the interval [ -3 , 3 ]:
[tex]f(x) = x^2+3x+4[/tex]
- The area "A1" bounded by f(x) is given as:
[tex]A1 = \int\limits^a_b {f(x)} \, dx[/tex]
Where, The interval of the function { -3 , 3 ] = [ a , b ]:
[tex]A1 = \int\limits^a_b {x^2+3x+4} \, dx\\\\A1 = \frac{x^3}{3} + \frac{3x^2}{2} + 4x |\limits_-_3^3 \\\\A1 = \frac{3^3}{3} + \frac{3*3^2}{2} + 4*3 - \frac{-3^3}{3} - \frac{3(-3)^2}{2} - 4(-3)\\\\A1 = 9 + 13.5 +12 + 9-13.5+12\\\\A1 =42 unit^2[/tex]
- Similarly for the other portion of piece-wise function covering the interval [3 , 4] :
[tex]f(x) = 4x+10[/tex]
- The area "A2" bounded by f(x) is given as:
[tex]A2 = \int\limits^a_b {f(x)} \, dx[/tex]
Where, The interval of the function { 3 , 4 ] = [ a , b ]:
[tex]A2 = \int\limits^a_b {4x+10} \, dx\\\\A2 = 2x^2 + 10x |\limits_3^4 \\\\A2 = 2*(4)^2 + 10*4 - 2*(3)^2 - 10*3\\\\A2 = 32 + 40 - 18-30\\\\A2 =26 unit^2[/tex]
- The total area "A" bounded by the piece-wise function over the entire domain [ -3 , 4 ] is given:
A = A1 + A2
A = 42 + 26
A = 68 unit^2
To find the area enclosed by f(x) and the x-axis from x = -3 to x = 4 for the given piecewise function, calculate the integral of each piece separately and then sum the areas. The area is divided into two parts due to the function having different expressions before and after x = 3.
Explanation:The student is asking to find the area under the curve of a given piecewise function f(x) on the interval from x = -3 to x = 4. Since the function is defined differently for x < 3 and x ≥ 3, the area calculation involves two parts:
Calculating the area under [tex]f(x) = x^2 + 3x + 4[/tex] from x = -3 to x = 3.Calculating the area under [tex]f(x) = 4x + 10[/tex] from x = 3 to x = 4.The first part can be calculated using the integral of [tex]f(x) = x^2 + 3x + 4[/tex]from x = -3 to x = 3. The second part is the integral of [tex]f(x) = 4x + 10[/tex] from x = 3 to x = 4. The total area is the sum of these two areas. For this function, the areas are bounded above by the function and below by the x-axis, so all areas are considered positive.
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Mikes milk shakes had a total of 9 gallons of ice cream they used 3 3/9 gallons this week and 1 1/3 gallons last week how many gallons of ice cream do they have left
Answer:
Step-by-step explanation:
[tex]3 \frac{3}{9}=\frac{3 \times 9+3}{9}=\frac{30}{9}=\frac{10}{3}\\1\frac{1}{3}=\frac{1\times 3+1}{3}=\frac{4}{3}\\Total~used=\frac{10}{3}+\frac{4}{3}=\frac{14}{3}\\Left~over=9-\frac{14}{3}=\frac{9 \times 3-14}{3}=\frac{13}{3}=4 \frac{1}{3}~gallons[/tex]
Answer: 4
Step-by-step explanation:
[tex]9-(3\frac{3}{9} + 1\frac{1}{3} )\\9-(\frac{30}{9} + \frac{4}{3})\\ 9-(\frac{10}{3}+ \frac{4}{3})\\ 9-(\frac{14}{3} )\\= \frac{(9)(3)-14}{3} \\= \frac{27-14}{3} \\=\frac{13}{3}\\ \\or\\\\4 gallons[/tex]
or
= [tex]4\frac{1}{3}[/tex]
approximately
okay so The top of the slide is 12 feet from the ground and has an angle of depression of 53 degree . What is the length of the slide? Round your answer to the nearest whole number. please help!!!
To find the length of the slide, you can use trigonometry and the angle of depression. The length can be found using the tangent function, and rounding to the nearest whole number gives an answer of approximately 9 feet.
Explanation:To find the length of the slide, we can use trigonometry and the angle of depression. We can use the tangent function, which is opposite over adjacent, to find the length of the slide. In this case, the opposite side is the height of the slide (12 feet) and the adjacent side is the length of the slide. So we have:
tan(53 degrees) = opposite/adjacent
tan(53 degrees) = 12/adjacent
To solve for the length of the slide, we can rearrange the equation:
adjacent = 12/tan(53 degrees)
Using a calculator, we can find that the length of the slide is approximately 9 feet when rounded to the nearest whole number.
Toms parents let him choose whether to play his favorite board games for 7/8 hours or 8/8 hours explain which amount of time you think tom should choose and why ?
Answer:
8/8
Step-by-step explanation:
8/8 is equal to one, so it would mean he would get to play his favorite board games for a full hour.
7/8 is only about 50 or so minutes, which is less than an hour.
If Tom chooses 8/8 of an hour, he will get to play longer.
A home decor store donated a percent of every sale to charity .The total sales were $7,400 so the store donated $148.What percent of $7,400 was donated to charity
The correct answer is 2%. The store donated 2% of $7,400 to charity.
To find out what percent of $7,400 was donated to charity, we need to divide the amount donated by the total sales and then multiply by 100 to convert it into a percentage.
Let's denote the percent donated as P. The amount donated is given as $148, and the total sales are $7,400. The relationship between the percent donated, the total sales, and the amount donated can be expressed as:
[tex]\[ P \times 7,400 = 148 \][/tex]
To find P, we divide both sides of the equation by 7,400:
[tex]\[ P = \frac{148}{7,400} \][/tex]
Now, we calculate the value of P:
[tex]\[ P = \frac{148}{7,400} = \frac{2}{100} \] \[ P = 0.02 \][/tex]
To express P as a percentage, we multiply by 100:
[tex]\[ P = 0.02 \times 100 \] \[ P = 2\% \][/tex]
In this question, all lengths are in cm. ABC. 3x-4, 2x+12, 7x-2. Given that AB:AC = 1:2 Show that AC:BC = 2:3
Answer: AC : BC = 22 : 33 = 2 : 3
Step-by-step explanation:
Since we have given that
AB = 3x-4
AC= 2x+12
BC = 7x-2
And AB: AC = 1 : 2
and To show : AC : BC = 2 : 3
So, it becomes ,
[tex]\dfrac{AB}{AC}=\dfrac{3x-4}{2x+12}=\dfrac{1}{2}\\\\2(3x-4)=2x+12\\\\6x-8=2x+12\\\\6x-2x=12+8\\\\4x=20\\\\x=\dfrac{20}{4}\\\\x=5[/tex]
So, AC: BC becomes:
[tex]\dfrac{AC}{BC}=\dfrac{3x-4}{7x-2}=\dfrac{2(5)+12}{7(5)-2}=\dfrac{22}{33}=\dfrac{2}{3}[/tex]
Hence, proved.
Step-by-step explanation:
First find x:
AB:AC=1:2
2(3x-4)=2x+12
6x-8=2x+12
4x-8=12
4x=20
x=5
Substitute:
AC=2x+12
=(2x5)+12
=10+12
=22
BC=7x-2
=(7x5)-2
=35-2
=33
22/33=2/3
Therefore AC:BC = 2:3
I need help with #1 and #2 how do I do this?
Answer:
2
Step-by-step explanation:
2
Answer:
The answer to your question is below
Step-by-step explanation:
Data
1) AB = 6 BC = 4 AE = 9 ED = ?
Use proportions to solve this problem
AB / AE = 6/9 (AB + BC) / (AE + ED) = (6 + 4) / (9 + ED)
-Equal both proportions
6/9 = 10 / (9 + ED)
-Solve for ED
9 + ED = 10(9)/ 6
-Simplification
9 + ED = 90/6
9 + ED = 15
ED = 15 - 9
-Result
ED = 6
2.- AB = 12 AC = 16 ED = 5 AE = ?
Use proportions to solve this problem
AE / AB = AE / 12 AD/AC = (AE + 5) / 16
-Equal both proportions
AE/12 = (AE + 5 ) / 16
-Solve for AE
16AE = 12(AE + 5)
16AE = 12AE + 60
16AE -12AE = 60
4AE = 60
AE = 60/4
-Result
AE = 15
Determine the length of arc JL. A) 25/ 9 π B) 5 /13 π C) 25 /18 π D) 14 /17 π
Answer:
Option C is the right choice where length of the arc is 25/18π.
Step-by-step explanation:
Given:
Length of the radius of the circle, [tex]r[/tex] = 2 unit
Angle between the arc, [tex]\theta[/tex] = 125°
We have to find the arc length.
Let the arc length JL be "a" unit.
Formula to be used:
Arc length = (Circumference times angle ) / 360°
Using the above formula and plugging the values.
⇒ Arc length, JL, 'a' = [tex]2\pi r\times (\frac{\theta}{360})[/tex]
⇒ [tex]a=2\pi r\times (\frac{\theta}{360})[/tex]
⇒ [tex]a=2\pi (2)\times (\frac{125}{360})[/tex]
⇒ [tex]a=4\pi \times (\frac{125}{360})[/tex]
⇒ [tex]a=\frac{4\pi \times 125}{360}[/tex]
⇒ [tex]a=\frac{500\pi }{360}[/tex]
⇒ [tex]a=\frac{50\pi }{36}[/tex]
⇒ [tex]a=\frac{25\pi }{18}[/tex] ... reducing to lowest term.
So,
The length of the arc JL is 25/18π and option C is the right choice.
The lower supports are and the area of the two supports is square meters. The upper arch can be decomposed as one semicircle with radius meters minus a semicircle with radius 3 meters. The area of the archway is (π + 24) square meters
Answer:
The lower supports are Congruent Rectangles and the area of the two supports is 24 square meters.
The upper arch can be decomposed as one semicircle with radius 6 meters minus a semicircle with radius 3 meters.
The area of the archway is (13.5 π + 24) square meters.
Step-by-step explanation:
See attachment for the firgure,
In order to determine the area of archway = The area of upper support + area of lower support.
As given, the lower support are two congruent rectangles consists of dimension 3 m × 4 m
Therefore, the area of lower support can be written as,
The area of lower support = 3 × 4 + 3 × 4 = 12 + 12 = 24 square m.
Now, the upper support arch can be decomposed as the two concentric semi circles having radius 3 m and 6 m,
Hence, the area of the upper support = Area of semi circle having radius i.e 6 m - Area of semi circle having radius i.e 3 m
=> π6²/2 - π3²/2= 27π/2= 13.5π square m
Therefore, the area of the archway = (13.5 π + 24) square meters
Answer:
1. congruent rectangles
2. 24
3. 6
4. 13.5
Step-by-step explanation:
Got it right, have a nice day!
Pete is conducting a survey to determine his customers’ overall satisfaction about the quality of his company’s products. He sends out surveys to the 5 customers who have purchased the largest number of items over the past year. Are his results likely to be representative of the population he is trying to analyze? Explain.
The sample is from the right population, his customers. However, because he is surveying such a small number of people and only asking his best customers (who are probably the most satisfied, or they would not buy so much from him), the results are going to be biased, most likely positive, and not representative of his entire customer base.
No, Because, when to do some sort of analysis (such as this one), you need to take (for example) a Random sample from the population of the problem that is being analyzed.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
Pete is conducting a survey to determine his customers’ overall satisfaction about the quality of his company’s products.
Let an example a Random sample from the population of the problem that is being analyzed.
In this example (problem), Pete wants to evaluate overall satisfaction of customers, so he should not send the surveys only to the customers who have purchased the largest number of items, but to the randomly selected customers, in order to obtain representative results of the overall satisfaction.
If he sends the surveys only to the customers who have bought the largest number of items, he will obtain very high satisfaction of customers, as results, of course, and this will not be representative results.
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work out the following questions
Answer:
work out the following questions
Step-by-step explanation:
a.
[tex] \frac{3}{4} \times \frac{1}{2} [/tex]
[tex] \frac{3}{8} [/tex]
Now
b.
[tex] \frac{4}{7} \times \frac{21}{40} [/tex]
Now here we can see 21 is 3 times of 7 and 40 is 10 times of 4 so
[tex] \frac{3}{10} [/tex]
Hope this helps.
An aquarium Is 8 feet long , 5feet wide , and 5.5 feet deep . What is the volume of the tank ?
Answer:
220 ft^3
Step-by-step explanation:
Assuming a rectangular tank,
V = l*w*h
= 8*5*5.5
=220 ft^3
PLSSSSSSSSSS HELLLPPPPPP!!! ASAP!!!! 6th GRADE MATH!!!!!!!!!! ITS FOR MY COUSIN!!!!!!FIRST TO ANSWER BOTH GETS MARKED BRAINLIEST!!!!
Answer:
I believe its 15
Step-by-step explanation:
Answer for 12+7+(-4)= 15
Answer for -1.75+(-6.25)+6= -2
I hope this helps
The distance traveled can be calculated using the formula D
= S. t where s is the speed and t is the total time. Which of
the situations below represent a total distance of 120 km?
Select all that apply.
A) 40 km/hr for 3 hours
B) 60 km/hr for 2 hours
C) 30 km/hr for 4 hours
D) 25 km/hr for 5 hours
Final answer:
Situations A (40 km/hr for 3 hours), B (60 km/hr for 2 hours), and C (30 km/hr for 4 hours) represent a total distance of 120 km when applying the formula D = S * t. Situation D (25 km/hr for 5 hours) does not represent the required distance of 120 km.
Explanation:
The distance traveled can be calculated using the formula D = S * t where S is the speed and t is the total time. To find out which situations represent a total distance of 120 km, we will apply this formula to each of the given situations.
A) 40 km/hr for 3 hours: D = 40 km/hr * 3 hr = 120 kmB) 60 km/hr for 2 hours: D = 60 km/hr * 2 hr = 120 kmC) 30 km/hr for 4 hours: D = 30 km/hr * 4 hr = 120 kmD) 25 km/hr for 5 hours: D = 25 km/hr * 5 hr = 125 kmAfter calculating each situation, we can see that situations A, B, and C each yield a total distance of 120 km, while situation D yields a total distance of 125 km, which is not the distance we are looking for.
Which expressions represent the area of the entire shaded
region, including the light and dark shading? Select three
options.
12a cm2
2b cm
(6a - b) cm
(12a + 2b) cm
(6a + b) cm
Answer:
12a cm2
2b cm2
(6a + b) cm2
Step-by-step explanation:
The complete question is
The figure consists of 12 congruent equilateral triangles. The area of one equilateral triangle is a cm2. The area of the hexagon, shaded slightly darker, is b cm2.
Which expressions represent the area of the entire shaded region, including the light and dark shading? Check all that apply.
12a cm2
2b cm2
(6a - b) cm2
(12a + 2b) cm2
(6a + b) cm2
The picture of the question in the attached figure
we know that
The area of the entire shaded region is equal to the area of the the light and dark shading
so
step 1
Find the area of the light shading
The area is equal to the area of six congruent equilateral triangles
[tex]A_1=6a\ cm^2[/tex]
step 2
Find the area of the dark shading
The area is equal to the area of the regular hexagon
[tex]A_2=b\ cm^2[/tex]
step 3
1) Find the total area
[tex]A=A_1+A_2=(6a+b)\ cm^2[/tex]
2) Remember that the figure consists of 12 congruent equilateral triangles
so
[tex]A=12a\ cm^2[/tex]
3) The area of the light shading is the same that the area of the dark shading
so
6a=b
therefore
[tex]A=2b\ cm^2[/tex]
Answer: 1 2 5
Step-by-step explanation:
Name: Joseph Sandoval, single, 1 exemption. Pay Rate: 5? per unit Unit produced: M-900, T-840, W-825, T-905, F-910
Answer:
1-4380 2-219 3-15.44 4-37.1 5-6.13 7-58.67 8-160.33
Step-by-step explanation:
Is 24, 45, and 51 a right triangle?
Answer: The dimensions are that of a right triangle.
Step-by-step explanation:
Answer:
yes
Step-by-step explanation:
you can solve by checking with the Pythagorean theorem (a^2 + b^2= c^2)
24 and 45 are the legs because the hypotenuse is the largest side so they will be substituted for a and b. if c is equal to 51 then it would be a right triangle
24^2 + 45^2= c^2
576 + 2025 = c^2
2601 = c^2
51 = c
because c = 51 and 51 was the hypotenuse on the triangle previously mentioned 24 45 51 is a right triangle
For two days in a row, Windstone rescued tadpoles from a puddle. He rescued 54 on Friday. This is 17 less Then the number he rescued on Saturday. Write and solve a subtraction equation to find how many tadpoles he rescued on Saturday.
Answer:
71 tadpoles
Step-by-step explanation:
Windstone rescued 54 on Friday.
Let the number rescued on Saturday =x
The number rescued on Friday is 17 less than the number he rescued on Saturday.
This is written as:
54=x-17To solve for x, we add 17 to both sides of the equation
54+17=x-17+1771=xx=71Windstone Rescued 71 tadpoles on Saturday
Answer:
71 tadpoles
Step-by-step explanation:
let 't' represent the number of tadpoles he rescued on Saturday.
According to the problem,
Windstone rescued 54 on Friday.
If the number rescued on Friday is 17 less than the number he rescued on Saturday.
The equation to find the number of tadpoles he rescued on Saturday will be:
t-17 = 54
Further solving for 't'
t= 54+17
t= 71
Therefore, Windstone Rescued 71 tadpoles on Saturday
Two angles in a triangle have measures of 23 and 98 what is the third measure of the third angle
Answer:
59
Step-by-step explanation:
As far as I know every triangle has to equal 180°
So if you add 98+23 you would get 121
subtract that from 180 and get 59
Hope this helps!
What is the mean for this data. Plz explain
Answer:
18
Step-by-step explanation:
Mean is the average of the data.
There is one 7.
Three 9's.
Two 10's
7+9+9+9+10+10=54
There are only 3 numbers with actual data so 6 and 8 don't matter.
54÷3=18
Trent began his math work at 5:29 p.m. He finished at 6:04 p.m. How many minutes did Trent spend on his math homework?
Answer: I am pretty sure it is 35 min
Step-by-step explanation:
Answer
Trent spent 35 minutes on his work.
Step-by-step explanation:
Eight less than twice a number is fourteen.
Answer:
x = 11
Step-by-step explanation:
2x - 8 = 14
2x = 22
x = 11
Tell me if I am wrong.
Can I get brainliest
Answer:
Let that number be 'x'
Now first part of the question tells us that 8 less than twice a number which is"x".
so twice a number would be 2x
then 8 less than would 2x-8
and the given result of the equation is 14
so the equation iis 2x-8=14
2x=14+8
2x=22
x=22/2
x=11 So the number is 11
Hope you find it helpful
PLZZZZZ HELP!!!!! WILL MARK BRAINLIEST. solve for t
–4(t − 12) + 12 = –4
t =
Answer:
t = 16
Step-by-step explanation:
–4(t − 12) + 12 = –4
-12 -12
-4(t - 12) = -16
/-4 /-4
t - 12 = 4
+12 +12
t = 16
or
–4(t − 12) + 12 = –4
distribute the -4
-4t + 48 + 12 = -4
simplify
-4t + 60 = -4
-60 -60
-4t = -64
/-4 /-4
t = 16
Answer:
64/3
Step-by-step explanation:
I need to find the surface area
Answer:
389.846
Step-by-step explanation
Sorry if its wrong
How are the graphs of the functions f(x) = V16 and g(x) = 3/64 related?
The functions f(x) and g(x) are equivalent.
The function g(x) increases at a faster rate.
The function g(x) has a greater initial value.
The function g(x) decreases at a faster rate.
Answer:
Step-by-step explanation:
hello :
* 64 = 4^3 so : g(x) = (∛4^3)^x = 4^x
* 16=4² so : f(x) = (√4²)^x = 4^x
the functions f(x) and g(x) are equivalent
If 5^3b-1 = 5^b-3 what is the value of b?
Answer:
b = -1
Step-by-step explanation:
Your equation is ...
(5^3)b -1 = 5^b -3
This is a mix of exponential and linear terms, and cannot be solved algebraically.
___
We suspect you might mean ...
5^(3b -1) = 5^(b -3)
We can equate the exponents to find a value for b:
3b -1 = b -3
2b = -2 . . . . add 1-b to both sides
b = -1 . . . . . . divide by 2
If x represents the amount Rita earns each week, which expression represents the amount she earns in a year?
x+52
52-
Write 1.7.7.7.7.7 using an exponent
Answer:
1^7 is the answer you are looking for
Step-by-step explanation:
Caitlyn finds that experimental probability of her making a three-point shot is 30%. Out of 500 three-point shots, about how many could she predict she would make?
Answer: 150 shots.
Step-by-step explanation: Since her three-point shot probably is 30%, we can assume that for any total of three-point shots, only 30% of them will be "correctly shot."
Here are two methods of how to find the answer, depending on how your teacher is teaching you to calculate percentages.
Method A. Proportions
Therefore, let's make a proportion. Out of 500 shots, only 30% will be made.
[tex]\frac{30}{100} =\frac{x}{500}[/tex]
Solve for x, which in a proportion, is done by cross-multiplying and then dividing.
[tex](30 * 500) / 100[/tex]
[tex]= 150[/tex]
Out of all the 500 shots, only 150 can be made by her prediction.
Method B. Simple multiplication
30% is equal to 0.3
Therefore, you multiply [tex]0.3 * 500[/tex], which you get [tex]150[/tex].
Calculate the portion of skiing in degrees Soccer - 5 hours Jogging - 20 Hours Walking - 10 hours Skiing - 5 hours Football - 10 hours Total - 50 hours
Answer: The portion of skiing is 36 degrees (36°)
Step-by-step explanation: In order to calculate the value of any of the given variables in degrees, we must calculate it as a fraction of the total and then convert it to a degree measure in relation to a total of 360 degrees. (Please note that a pie chart has a total of 360 degrees measurement for all the variables given)
The variables are;
Soccer - 5
Jogging - 20
Walking - 10
Skiing - 5
Football - 10
Total - 50
Therefore, the fraction that represents skiing is given as
Skiing = 5/50
Skiing = 1/10
Calculating this portion in degrees becomes,
Skiing = 1/10 x 360
Skiing = 36 degrees (36°)
Answer:
x = 36° represents the proportion of hours spent on skiing.
Step-by-step explanation:
Solution:-
- The time taken for various activities is to be represented using a pie chart.
- The individual times for various activities are given:
Soccer : 5 hours
Jogging : 20 hours
Walking : 10 hours
Skiing : 5 hours
Football : 10 hours
=========================
Total time : 50 hours
=========================
- An entire pie chart is represented by a circle. Where the angle representation of a complete circle is 360° - mapping the total number of hours taken for all activities.
- So, 360° ............. Total Time = 50 hours
- So we will use direct proportions to evaluate the angle representation for the number of hours spent on skiing for a pie chart:
x ................. Skiing time = 5 hours
======================================
50*x = 5*360
x = 36°
What is the volume of this rectangular prism? 6/5cm 8/3cm 15/4cm
Explanation of how to calculate the volume of a rectangular prism with given dimensions.
Explanation:The volume of a rectangular prism is calculated by multiplying its length, width, and height.
Given dimensions: length = 6/5 cm, width = 8/3 cm, height = 15/4 cm
Volume = (6/5) cm x (8/3) cm x (15/4) cm = 24 cm³