4. A pool measuring 24 feet by 16 feet is
surrounded by a uniform path of width x feet.
The total enclosed area is 768 ft².
Find x, the width of the path.
The width of the path, x, is 48 feet
How to determine the parametersThe formula for determining the area of a rectangle is expressed as;
Area = lw
Where;
l is the length of the given rectanglew is the width of the given rectangleFrom the image shown and the information given, we can see that;
The width is given as = x
The area of the rectangle = 768 ft²
The length of the rectangle = 16
Now, substitute the values, we have;
768 = 16x
Make 'x' the subject of formula by dividing both sides by its coefficient, we have;
768/16 = 16x/16
Find the quotient
x = 48 feet
But, we have;
Width = x = 48 feet
Hence, the value is 48 feet
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find the first second and third derivatives of the function
Given the function
[tex]f(x)=\frac{8}{5}x-9[/tex]Finding the derivative we have
[tex]f^{^{\prime}}(x)=\frac{8}{5}[/tex]Also
[tex]f^{\doubleprime}(x)=0^{}[/tex]Finally
[tex]f^{^{\doubleprime}^{\prime}}(x)=0[/tex]The floor of a shed has an area of 80 square feet. The floor is in the shape of a rectangle whose length is 6 feet less than twice the width. Find the length and the width of the floor of the shed. use the formula, area= length× width The width of the floor of the shed is____ ft.
Given:
The area of the rectangular floor is, A = 8- square feet.
The length of the rectangular floor is 6 feet less than twice the width.
The objective is to find the measure of length and breadth of the floor.
Consider the width of the rectangular floor as w, then twice the width is 2w.
Since, the length is given as 6 feet less than twice the width. The length can be represented as,
[tex]l=2w-6[/tex]The general formula of area of a rectangle is,
[tex]A=l\times w[/tex]By substituting the values of length l and width w, we get,
[tex]\begin{gathered} 80=(2w-6)\times w \\ 80=2w^2-6w \\ 2w^2-6w-80=0 \end{gathered}[/tex]On factorizinng the above equation,
[tex]\begin{gathered} 2w^2-16w+10w-80=0 \\ 2w(w-8)+10(w-8)=0 \\ (2w+10)(w-8)=0 \end{gathered}[/tex]On solving the above equation,
[tex]\begin{gathered} 2w+10=0 \\ 2w=-10 \\ w=\frac{-10}{2} \\ w=-5 \end{gathered}[/tex]Similarly,
[tex]\begin{gathered} w-8=0 \\ w=8 \end{gathered}[/tex]Since, the magnitude of a side cannot be negative. So take the value of width of the rectangle as 8 feet.
Substitute the value of w in area formula to find length l.
[tex]\begin{gathered} A=l\times w \\ 80=l\times8 \\ l=\frac{80}{8} \\ l=10\text{ f}eet. \end{gathered}[/tex]Hence, the width of the floor of the shed is 8 ft.
Question 5 Fill in the table. First Integer Next Integers Give four consecutive odd integers: The simplified sum of the second and forth integers are Question Help: Message instructor Submit Question
The four consecutive odd integers
If the first integer is given to be x
Then the next three are:
x + 2, x+ 4 and x+ 6
The sum of the second and forth integers :
x+2 + x+ 6 = 2x + 8
Hence, the sum of the second and forth integers are: 2x+8
A carpenter wants to cut a board that is 5/6 ft long into pieces that are 5/16 ft long. The carpenter will use the expression shown to calculate the number of pieces that can be cut from the board.5/6 divided by 5/16How many pieces can be cut from the board?
The expression which is used to calculate the number of pieces that can be cut from the board is:
[tex]\frac{5}{6}\div\frac{5}{16}[/tex]We solve this by changing the division sign to multiplication and taking the reciprocal of the second fraction.
Therefore:
[tex]\begin{gathered} \frac{5}{6}\div\frac{5}{16}=\frac{5}{6}\times\frac{16}{5} \\ =\frac{16}{6} \\ =2\text{ }\frac{4}{6} \\ =2\frac{2}{3}\text{ pieces} \end{gathered}[/tex]The carpenter can cut 2 2/3 pieces from the board.
Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4m×3m ?
Given:
Length = 4m
Width= 3m
Height = 2.5 m
Therefore, the surface area of rectangle prism is 2lh+2bh+lb
[tex]\begin{gathered} 4\times2.5\times2+3\times2.5\times2+4\times3=10\times2+5\times3+12 \\ =20+15+12 \\ =47 \end{gathered}[/tex]Hence, the required answer is 47m^2.
Question 8 Let h(t) = –1612 +64 + 80 represent the height of an object
To find the time it takes the object to reach the maximum height we need to remember that this happens in the axis of symmetry of the parabola described by the function:
[tex]h(t)=at^2+bt+c[/tex]The axis of symmetry is given as:
[tex]t=-\frac{b}{2a}[/tex]in this case we have that a=-16 and b=64, then we have:
[tex]t=-\frac{64}{2(-16)}=\frac{-64}{-32}=2[/tex]Therefore it takes 2 seconds to the object to reach its maximum height.
Now, to find the maximum height we plug this value of t in the equation, then we have:
[tex]\begin{gathered} h(2)=-16(2)^2+64(2)+80 \\ =-16(4)+128+80 \\ =-64+128+80 \\ =144 \end{gathered}[/tex]therefore the maximum height is 144 ft.
Which of the sketches presented in the list of options is a reasonable graph of y = |x − 1|?
ANSWER
EXPLANATION
The parent function is y = |x|. The vertex of this function is at the origin.
When we add/subtract a constant from the variable, x, we have a horizontal translation, so the answer must be one of the first two options.
Since the constant is being subtracted from the variable, the translation is to the right. Hence, the graph of the function is the one with the vertex at (1, 0).
which answer choice gives the correct surface area for a triangular prism with bases that are 4 cm2 and sides that are 10 cm2? A. 12 cm2 B. 26 cm2 C.38 cm2 D. 40 cm2
Explanation
A trinagular prism has two bass and theree side surfaces.
Therefore, the suface area pf the prism is
[tex]S.A=3(10)+2(4)=30+8=38cm^2[/tex]Answer: Option C
Which of the following tables shows a uniform probability model?
The answer is the third choice
Where all probability are equal
Give. ∆ABC Angle B = 42°, Angle C = 71° and BC = 22. Find AB and round your answer to nearest integer.
Let's make a diagram to visualize the problem.
First, let's find angle A.
[tex]\begin{gathered} A+B+C=180 \\ A+42+71=180 \\ A=180-71-42 \\ A=67 \end{gathered}[/tex]Then, we use the law of sines to find AB.
[tex]\begin{gathered} \frac{AB}{\sin71}=\frac{BC}{\sin A} \\ \frac{AB}{\sin71}=\frac{22}{\sin 67} \\ AB=\frac{22\cdot\sin 71}{\sin 67} \\ AB\approx23 \end{gathered}[/tex]Therefore, AB is 23 units long, approximately.Question 8 of 10If f(x) = - VX-3, complete the following statement (round your answerto the nearest hundredth):3x + 2f(7) = —Answer hereSUBMITplease help
To find f(7) substitute x by 7 in the function
Hi, The area of a circle is 100 quare millimeters. The radius is 5.64 millimeters. what is the circumference?
The area A of a circle is given by
[tex]A=\pi r^2[/tex]where Pi is 3.1416 and r is the radius. In our case, we get
[tex]100\operatorname{mm}=\pi r^2[/tex]and we need to find r. In this regard, if we move Pi to the left hand side we get
[tex]\frac{100}{\pi}=r^2[/tex]then, r is given by
[tex]r=\sqrt[]{\frac{100}{\pi}}[/tex]Now, the circunference C is given by
[tex]C=2\pi\text{ r}[/tex]then, by substituting our last result into this formula, we have
[tex]C=2\pi\sqrt[]{\frac{100}{\pi}}[/tex]since square root of 100 is 10, we get
[tex]C=2\pi\frac{10}{\sqrt[]{\pi}}[/tex]we can rewrite this result as
[tex]\begin{gathered} C=\frac{2\pi\times10}{\sqrt[]{\pi}} \\ C=\frac{2\sqrt[]{\pi\text{ }}\sqrt[]{\pi}\times10}{\sqrt[]{\pi}} \end{gathered}[/tex]and we can cancel out a square root of Pi. Then, we have
[tex]C=2\sqrt[]{\pi}\times10[/tex]and the circunference is
[tex]C=20\text{ }\sqrt[]{\pi}\text{ milimeters}[/tex]Which system of equations best represents the situation below?A farmer grew his own tomatoes (a), eggplants (b), and potatoes (c). Hedecided to package his vegetables and price them as follows:1 tomato, 1 eggplant, 2 potatoes for $102 tomatoes, 1 eggplant, 3 potatoes for $144 tomatoes, 3 eggplants, 5 potatoes for $20
Solution
- The cost of the crops are $(a) for tomatoes, $(b) for eggplants, and $(c) for potatoes.
- We simply need to follow the statements about the farmer's pricing in order to determine the correct set of equations.
Statement 1:
- "1 tomato, 1 eggplant, 2 potatoes for $10"
- If there is 1 tomato, it implies that, this tomato is priced at $(a). Similarly, 1 eggplant would be priced at $(b), but 2 potatoes would be $(c) + $(c) = $2(c).
- We are told that the total cost for this package is $10.
- Thus, the first equation must be:
[tex]a+b+2c=10[/tex]- We can interprete the other packages in a similar manner.
Statement 2:
"2 tomatoes, 1 eggplant, 3 potatoes for $14"
- This implies that the farmer would price the packages as follows:
2 tomatoes: 2(a)
1 eggplant: 1(b)
3 potatoes: 3(c)
- Since the total cost is $14, we can write the second equation as follows:
[tex]2a+b+3c=14[/tex]Statement 3:
"4 tomatoes, 3 eggplants, 5 potatoes for $20"
- This implies that the farmer would price the packages as follows:
4 tomatoes: 4(a)
3 eggplants: 3(b)
5 potatoes: 5(c)
- Since the total cost is $20, we can write the third equation as follows:
[tex]4a+3b+5c=20[/tex]Final Answer
The 3 equations are:
[tex]\begin{gathered} a+b+2c=10 \\ 2a+b+3c=14 \\ 4a+3b+5c=20 \end{gathered}[/tex]OPTION C
help meeeee pleaseeeee!!!
thank you
Answer:
(f o g) = 464
Step-by-step explanation:
f(x) = x² - 3x + 4; g(x) = -5x
(f o g)(4) = f(g(4))
f(g(4)) = -5(4) = -20
f(g(4)) = (-20)² - 3(-20) + 4
f(g(4)) = 464
I hope this helps!
Find the midpoint for the line segment whose endpoints are (-10,11) and (-1,-15).
Answer:
( -11/2, -2)
Step-by-step explanation:
Finding the midpoint
To find the x coordinate of the midpoint, add the x coordinates of the endpoints and then divide by 2
(-10+-1)/2 = -11/2
To find the y coordinate of the midpoint, add the y coordinates of the endpoints and then divide by 2
(11+-15)/2 = -4/2 = -2
The mid point is ( -11/2, -2)
How far is the bottom of the ladder from thebottom of the wall? Use the PythagoreanTheorem to determine the solution. Explain howyou found your answer.
The Pythagorean Theorem is
[tex]c^2=a^2+b^2[/tex]where
c=hypotenuse=13
a=12
b=x
then we substitute the values
[tex]13^2=12^2+x^2[/tex]then we isolate the x
[tex]\begin{gathered} x=\sqrt[]{13^2-12^2} \\ x=\sqrt[]{169-144} \\ x=\sqrt[]{25} \\ x=5 \end{gathered}[/tex]The bottom of the ladder is 5m far from the bottom of the wall
The path of a race will be drawn on a coordinate grid like the one shown below. The starting point of the race will be at (-5.3, 1). The finishing point will be at(1, -5.3). Quadranto Quadrant P Quadrant Quadrants Part A: Use the grid to determine in which quadrants the starting point and the finishing point are located. Explain how you determined the locations. (6 points) Part B: A checkpoint will be at (5.3, 1). In at least two sentences, describe the difference between the coordinates of the starting point and the checkpoint, and explain how the points are d. (4 points)
The path of a race will be drawn on a coordinate grid like the one shown below. The starting point of the race will be at (-5.3, 1). The finishing point will be at(1, -5.3). Quadranto Quadrant P Quadrant Quadrants Part A: Use the grid to determine in which quadrants the starting point and the finishing point are located. Explain how you determined the locations. (6 points) Part B: A checkpoint will be at (5.3, 1). In at least two sentences, describe the difference between the coordinates of the starting point and the checkpoint, and explain how the points are d. (4 points)
Part A
we have
starting point of the race is (-5.3, 1)
the x-coordinate is negative and the y coordinate is ;positive
that means-------> is located on quadrant Q
finishing point is (5,3, 1)
x-coordinate is postive and y coordinate is positive
that means -----> is located on Quadrant P
Answer:
Step-by-step explanation:
part A
x-y=3x+y=5unit 7 systems of linear equations
then
[tex]\begin{gathered} x+y=5 \\ 3+y+y=5 \\ 3+2y=5 \\ 3+2y-3=5-3 \\ 2y=2 \\ \frac{2y}{2}=\frac{2}{2} \\ y=1 \end{gathered}[/tex]solve for x
[tex]\begin{gathered} x=3+y \\ x=3+1 \\ x=4 \end{gathered}[/tex]answer: C. (4,1)
Find the x - and y -intercepts of the graph of the linear equation -6x + 9y = -18
Someone else got x=(3,0) y=(0,-2) but it was wrong
Answer:
x-intercept = 3y-intercept = -2Step-by-step explanation:
You want the intercepts of the equation -6x +9y = -18.
InterceptsThere are several ways to find the intercepts. In each case, the x-intercept is the value of x that satisfies the equation when y=0, and vice versa.
For y = 0, we find the x-intercept to be ...
-6x + 0 = -18
x = -18/-6 = 3
The x-intercept is 3; the point at that intercept is (3, 0).
For x = 0, we find the y-intercept to be ...
0 +9y = -18
y = -18/9 = -2
The y-intercept is -2; the point at that intercept is (0, -2).
Intercept formThe intercept form of the equation for a line is ...
x/a +y/b = 1
where 'a' is the x-intercept, and 'b' is the y-intercept.
We can get this form by dividing the original equation by -18.
-6x/-18 +9y/-18 = 1
x/3 +y/(-2) = 1
The x-intercept is 3; the y-intercept is -2.
__
Additional comment
When asked for the intercepts, it is sometimes not clear whether you are being asked for the value where the curve crosses the axis, or whether you are being asked for the coordinates of the point there.
Your previous "wrong" answer was given as point coordinates. Apparently, just the value at the axis crossing is required.
You have to have some understanding of your answer-entry and answer-checking software to tell the required form of the answer (or you can ask your teacher).
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Write the decimal as a quotient of two integers in reduced form.
0.513
The given decimal can be written as a quotient of 513/1000.
What is quotient?
In maths, the result of dividing a number by any divisor is known as the quotient. It refers to how many times the dividend contains the divisor. The statement of division, which identifies the dividend, quotient, and divisor, is shown in the accompanying figure. The dividend 12 contains the divisor 2 six times. The quotient is always less than the dividend, whether it is larger or smaller than the divisor.
we can write the decimal given 0.513 as a answer of of 513 divided by 1000.
I.e.
[tex]0.513 = \frac{513}{1000}[/tex]
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SOMEONE PLS HELPPPPPPPP
Answer:
**NEED USEFUL ANSWER ASAP, H.W QUESTION**
Given that hotter blackbodies produce more energy than cooler blackbodies, why do cooler red giants have much higher luminosities than much hotter white dwarfs?
Step-by-step explanation:
12x÷4yif x=-8 and y=3
To solve 12x÷4y, first, let's evaluate the products on both sides of the ÷ symbol, we know that x = -8, then we have:
[tex]12\times(-8)=-96[/tex]We have -96 on the left side of the ÷ symbol.
We know that y = 3, then, on the right side, we have:
[tex]4\times3=12[/tex]Then, we have 12 on the right side of the ÷ symbol, now the expression looks like this:
-96 ÷ 12. what we have to do is to divide -96 by 12, then we get:
[tex]-96\text{ }\div12=\frac{-96}{12}=-8[/tex]Then, the answer is 8
Which points are separated by a distance of 4 units?A. (3,6) (3,9)B. (2,7) (2,3)C. (1,5) (1,3)D. (4,2) (4,7)
let us take the point (2,7) and (2,3) the distance between these two is
[tex]\begin{gathered} d=\sqrt[]{(2-2)^2+(7-3)^2} \\ d=\sqrt[]{4^2} \\ d=4\text{ unit} \end{gathered}[/tex]Hence these two points are separated by 4 units.
So option B is correct.
Hi, could I have some help answering this question in the picture attached?simplify the question
Expand and collect like terms:
[tex]\begin{gathered} =\text{ }7s^{\frac{7}{4}}\times t^{\frac{-5}{3}}\times-6s^{\frac{-11}{4}}\times t^{\frac{7}{3}} \\ =\text{ }7\times s^{\frac{7}{4}}\times-6\times s^{\frac{-11}{4}}\times t^{\frac{-5}{3}}\times t^{\frac{7}{3}} \\ =\text{ 7 }\times-6\text{ }\times\text{ }s^{\frac{7}{4}}\times s^{\frac{-11}{4}}\times t^{\frac{-5}{3}}\times t^{\frac{7}{3}} \\ =\text{ -42}\times\text{ }s^{\frac{7}{4}}\times s^{\frac{-11}{4}}\times t^{\frac{-5}{3}}\times t^{\frac{7}{3}} \end{gathered}[/tex]Bring the exponents having same base together:
[tex]\begin{gathered} \text{The multiplication betwe}en\text{ same base becomes addition } \\ \text{when the exponents are brought together} \\ =-42\text{ }\times\text{ }s^{\frac{7}{4}-\frac{11}{4}}\times t^{\frac{-5}{3}+\frac{7}{3}} \\ =\text{ -42 }\times s^{\frac{7-11}{4}}\times t^{\frac{-5+7}{3}} \\ =\text{ -42 }\times s^{\frac{-4}{4}}\times t^{\frac{2}{3}} \end{gathered}[/tex][tex]\begin{gathered} =\text{ -42 }\times s^{\frac{-4}{4}}\times t^{\frac{2}{3}} \\ =\text{ -42 }\times s^{-1}\times t^{\frac{2}{3}} \\ =\text{ -42}s^{-1}t^{\frac{2}{3}} \end{gathered}[/tex]Describe the transformation of f(x) that produce g(x). f(x)= 2x; g(x)= 2x/3+7Choose the correct answer below.
The vertical translation involves shifting the graph either up or down on the y axis. For example.
[tex]\begin{gathered} y=f(x) \\ \text{translated upward }it\text{ will be } \\ y=f(x)+k \end{gathered}[/tex]When a graph is vertically compressed by a scale factor of 1/3, the graph is also compressed by that scale factor. This implies vertical compression occurs when the function is multiplied by the scale factor. Therefore,
[tex]\begin{gathered} f(x)=2x \\ \text{The vertical compression by a scale of }\frac{1}{3}\text{ will be} \\ g(x)=\frac{1}{3}(2x)=\frac{2}{3}x \end{gathered}[/tex]Finally, the vertical translation up 7 units will be as follows
[tex]g(x)=\frac{2}{3}x+7[/tex]The answer is a. There is a vertical compression by a factor of 1/3 . Then there is a vertical translation up 7 units.
Find the 11th term of the arithmetic sequence -5x- 1, -8x + 4, -11 x+ 9, ...
Recall that an arithmetic sequence is a sequence in which the next term is obtained by adding a constant term to the previous one. Let us consider a1 = -5x-1 as the first term and let d be the constant term that is added to get the next term of the sequence. Using this, we get that
[tex]a_2=a_1+d[/tex]so if we replace the values, we get that
[tex]-8x+4=-5x-1+d[/tex]so, by adding 5x+1 on both sides, we get
[tex]d=-8x+4+5x+1\text{ =(-8+5)x+5=-3x+5}[/tex]To check if this value of d is correct, lets add d to a2. We should get a3.
Note that
[tex]a_2+d=-8x+4+(-3x+5)=-11x+9=a_3[/tex]so the value of d is indeed correct.
Now, note the following
[tex]a_3=a_2+d=(a_1+d)+d=a_1+2d=a_1+d\cdot(3-1)[/tex]This suggest the following formula
[tex]a_n=a_1+d\cdot(n-1)[/tex]the question is asking for the 11th term of the sequence, that is, to replace the value of n=11 in this equation, so we get
[tex]a_{11}=a_1+d\cdot(10)=-5x-1+10\cdot(-3x+5)\text{ =-5x-1-30x+50 = -35x+49}[/tex]so the 11th term of the sequence is -35x+49
PLS HELP Quadrilateral ABCD is located at A(-2, 2), B(-2, 4), C(2, 4), and D(2, 2). The quadrilateral is then transformed using the rule (x + 7, y - 1) to form the imagecoordinates of A', B', C', and D'? Describe what characteristics you would find if the corresponding vertices were connected with line segments
Given:
The coordinates of Quadrilateral ABCD is A(-2, 2), B(-2, 4), C(2, 4), and D(2, 2).
The quadrilateral is transformed with the rule,
[tex](x,y)\rightarrow\mleft(x+7,y-1\mright)[/tex]It becomes,
[tex]\begin{gathered} A\mleft(-2,2\mright)\rightarrow A^{\prime}\mleft(-2+7,2-1\mright)=A^{\prime}(5,1) \\ B\mleft(-2,4\mright)\rightarrow B^{\prime}(-2+7,4-1)=B^{\prime}(5,3) \\ C\mleft(2,4\mright)\rightarrow C^{\prime}(2+7,4-1)=C^{\prime}(9,3) \\ D(2,2)\rightarrow D^{\prime}(2+7,2-1)=D^{\prime}(9,1) \end{gathered}[/tex]Now, join the corresponding vertices of both the quadrilateral with the line segment.
After joining the vertices of the quadrilateral ABCD and A'B'C'D'. it gives the 3-dimensional shape- a rectangular prism.
3. Suppose an investment of $5000 doubles every 12 years. How much is the investment worth after: 24 years?
Money = $5000
time = 12 years
investment after 24 years
If the investment doubles every 12 years after 24 years the total amount of money will be $10000.0
The length of a room is twice as its breadth and breadth is 6 cm. If it's height is 4 cm, find the total surface area.
The breadth of the room = 6 cm
Since the length of the room is twice its breadth
Then
Length of the room = 2 times 6cm = 12cm
The height of the room = 4cm
Since the shape of the room is a cuboid
The surface area of a cuboid is given as
[tex]SA=2(lh+lw+hw)[/tex]Substitute l = 12, w = 6 and h = 4 into the formula
This gives
[tex]SA=2(12\times4+12\times6+4\times6)_{}[/tex]Simplify the expression
[tex]\begin{gathered} SA=2(48+72+24) \\ SA=2(144) \\ SA=288 \end{gathered}[/tex]Therefore, the total surface area of the room is
[tex]288cm^2[/tex]