• An ice cube is slowly melting, losing 3cm^3 of water each hour. If it is always a perfect cube, (V=s^3), what is the rate of change of its side length when it has 8 cm^3 of ice left?
Answers
Given:
The volume is decreasing at the rate of 3 cm^3 per hour.
The volume of the left ice is 8 cm^3.
Aim:
We need to find the rate of change of the side of the cube.
Explanation:
Let the length of the cube is denoted as s.
Consider the volume of the cube.
[tex]V=s^3[/tex]Since the volume is decreasing at the rate of 3 cm^3 per hour. we can write,
[tex]\frac{dV}{dt}=-3cm^3\/h[/tex]where t represents time and the negative sign represents decreasing.
Differentiate the volume with respect to s.
[tex]\frac{dV}{ds}=\frac{d}{ds}(s^3)=3s^2[/tex]To find the rate of change of the side length, we use the chain rule.
[tex]\frac{dV}{dt}=\frac{dV}{ds}\frac{ds}{dt}[/tex][tex]\text{ Substitute }\frac{dV}{dt}=-3\text{ and }\frac{dV}{ds}=3s^2\text{ in the equation.}[/tex][tex]-3=\frac{ds}{dt}(3s^2)[/tex][tex]-\frac{3}{3s^2}=\frac{ds}{dt}[/tex][tex]-\frac{1}{s^2}=\frac{ds}{dt}[/tex]Since the left ice is 8 cm ^3.
[tex]V=(s)^3=8[/tex][tex]s^3=2^3[/tex][tex]s=2cm[/tex][tex]Substitute\text{ s =2 in the equation}-\frac{1}{s^2}=\frac{ds}{dt}.[/tex][tex]-\frac{1}{2^2}=\frac{ds}{dt}.[/tex][tex]\frac{ds}{dt}=-\frac{1}{4}[/tex][tex]\frac{ds}{dt}=-0.25cm\text{ per hour}[/tex]Verification:
Let s =2 cm, then the volume is 8cm^3.
Let s =1.75cm, the volume is
Related Questions
Geometry Question: Given segment EA is parallel segment DB, segment EA is congruent to segment DB, and B is the mid of segment AC; Prove: segment EB is parallel to segment DC (reference diagram in picture)
Answers
Construction: Join ED.
The corresponding diagram is given below,
According to the given problem,
[tex]\begin{gathered} AE=BD \\ AE\parallel BD \end{gathered}[/tex]Since a pair of opposite sides are parallel and equal, it can be claimed that quadrilateral ABDE is a parallelogram.
Then, as a property of any parallelogram, it can be argued that,
[tex]\begin{gathered} AB=DE \\ AB\parallel DE \end{gathered}[/tex]Given that B is the mid-point of AC,
[tex]\begin{gathered} AB=BC \\ AB\parallel BC \end{gathered}[/tex]Combining the above two results,
[tex]\begin{gathered} BC=DE \\ BC\parallel DE \end{gathered}[/tex]It follows that ABCD also forms a parallelogram.
Again using the property that opposite sides of a parallelogram are equal and parallel. It can be claimed that,
[tex]\begin{gathered} EB=DC \\ EB\parallel DC \end{gathered}[/tex]Hence proved that segment EB is parallel to segment DC,
[tex]\vec{EB}=\vec{DC}[/tex]For each value of w, determine whether it is a solution to w < 9.Is it a solution?W5?YesNo75914
Answers
Answer: 5 and 7
Explanation:
we need to determine if a number is a solution to
[tex]w<9[/tex]That reads as "w is less than 9, and not equal to 9"
so we find in our options wich ones are less than 9. The options are:
• 7
,• 5
,• 9
,• 14
The ones smaller or less than 9 are: 5 and 7
The ones greater than 9 or equal to 9 (the ones that are not the solution) are: 9 and 14.
So the solutions are: 5 and 7
I need the answer to number 2 please answer it like the paper so that I can understand it better. Please
Answers
Elijah made an error of subtracting the numbers and not including the imaginary sign (letter i)
The correct midpoint is (6, 3i)
Explanation:The two points are 8 + 4i and 4 + 2i
Elijah got the midpoint as (2, 1).
To determine Elijah's error, let's calculate the midpoint of a complex number:
[tex]\text{Midpoint = (}\frac{a\text{ + b}}{2}),\text{ (}\frac{c\text{ + d}}{2})i[/tex]let 8 + 4i = a + ci
let 4 + 2i = b + di
The real numbers will be added together. The imaginary numbers will also be added together.
substituting the values in the formula:
[tex]\begin{gathered} \text{Midpoint = (}\frac{8\text{ + 4}}{2}),\text{ (}\frac{4\text{ + 2}}{2})i \\ \text{Midpoint = }\frac{12}{2},\text{ }\frac{6}{2}i \\ \\ \text{Midpoint = (6, 3i)} \end{gathered}[/tex]Elijah made an error of subtracting the numbers. Also Elijah didn't include the letter representing the imaginary numbers (i).
The correct midpoint is (6, 3i)
Calculate the value of the expression 3x-7 when x = 2
Answers
Given:
The expression is,
[tex]3x-7[/tex]To find:
The value when x = 2.
Explanation:
Substitute x = 2 in the given expression, we get
[tex]\begin{gathered} 3(2)-7=6-7 \\ =-1 \end{gathered}[/tex]Thus, the value of the expression when x = 2 is -1.
Final answer:
The value of the expression when x = 2 is,
[tex]-1[/tex]ASGC is also considering adding tennis racquets to the product lines it produces. This would require a $500,000 modification to its factory as well as the purchase of new equipment that costs $1,600,000. The variable cost to produce a tennis racquet would be $55, but John thinks that ASGC could sell the racquet at a wholesale price of $75. John thinks that if ASGC sells the racquet at a lower price, many other retailers might decide to carry it. However, the vice president of ASGC thinks that the tennis racquet is a superior product and that ASGC should sell it for $99.99 to upscale country clubs only. The higher price would give a prestige image. Questions based on the above (10 pts)7. If ASGC produces tennis racquets, how many racquets must it sell at $75.00 and $99.99 to break even? •Breakeven units at 75.00 _______________________________. •Breakeven units at 99.99 _______________________________. •Which price do you recommend and why? __________________________
Answers
Solution
[tex]undefined[/tex]how many minutes until the heart beats 200 times
Answers
From the given table, we can read that the 200 beats is associated with the entry: "5 minutes", so that is the answer we pick.
which agrees with the first option in the provided list of possible answers.
Can you please help me out with a question
Answers
the figure is composed by a 4 triangles and a cube
to find the area of a triangle we need the base and height. the base is 15ft
to find the height we mut use the pithagorean theorem
h= height of the traingle
[tex]h^2=(15ft)^2+(7.5ft)^2[/tex]resolving we have
[tex]h=\sqrt[\square]{281.25}\text{ = 16.78 aprox}[/tex]and now he have all the measures
each triangle at the top has an area equal to
[tex]A=\frac{16.78ft\cdot15ft}{2}=127.78sq\text{ ft}[/tex]now we multiply that by 4: 127.78sq ft*4=503.1 sq ft
for the bottom part, there are 5 squares of side 15ft
each square has an area = 15ft*15ft = 225 sq ft
multipliying that by 5: 225sqft*5=1125 sq ft
the total area is 1125 sq ft+503.1sqft=1628.1 sq ft rounded is 1628 sq ft
For the volume of the piramid, we use
[tex]V=\frac{1}{3}A\cdot h[/tex]where A is the area of the base and h is the height
so volume of piramid:
[tex]V=\frac{1}{3}\cdot225\text{sqft}\cdot15ft=1125ft^3[/tex]for the volume of the cube we multiply the side length 3 times:
[tex]V\mleft(cube\mright)=(15ft)^3=3375ft^3[/tex]Adding the two volumes:
1125ft^3+3375ft^3=4500 cubic feet
list all numbers from the given set that are
Answers
Part a
Natural numbers are:
[tex]\sqrt[]{25}[/tex]because
[tex]\sqrt[]{25}=5[/tex]Part b
whole numbers
[tex]0,\text{ }\sqrt[]{25}[/tex]Part c
Integers
[tex]-9,0,\sqrt[]{25}[/tex]Part d
rational numbers
[tex]\frac{3}{4},-9,0.6,0,8.5,\sqrt[]{25}[/tex]Part e
Irrational numbers
[tex]\pi,\text{ }-\sqrt[]{2}[/tex]Part f
real numbers
[tex]\frac{3}{4},-9,0.6,0,\pi,8.5,\sqrt[]{25},\text{ -}\sqrt[]{2}[/tex]Find the z-score location of a vertical line that separates anormal distribution as described in each of the following.a. 15% in the tail on the rightb. 40% in the tail on the leftc. 75% in the body on the rightd. 60% in the body on the left
Answers
Answer:
a. z = 1.0364
b. z = -0.2533
c. z = -0.6745
d. z = 0.2533
Explanation:
We can represent each option with the following diagrams
So, for each option, we need to find a z that satisfies the following
a. P(Z > z) = 0.15
b. P(Z < z) = 0.40
c. P(Z > z) = 0.75
d. P(Z > z) = 0.60
Then, using a normal table distribution, we get that each value of z is
a. z = 1.0364
b. z = -0.2533
c. z = -0.6745
d. z = 0.2533
The graph of an inequality has a closed circle at 4.3, and the ray moves to the right. What inequality is graphed?x > 4.3x ≥ 4.3x ≤ 4.3x < 4.3
Answers
From the question, we were told that the graph of an inequality has a closed circle at 4.3 and the ray moves to the right also.
We are to determined the inequality that is graphed from the options.
From what is seen, we want x to be greater than or equal to 4.3.
The closed circle tells us that it can be equal to 4.3. The ray to the right tells us that we are looking for numbers larger than 4.3.
So the inequality graphed is that of x is greater than or equal to 4.3
So the correct option is the second option which is x ≥ 4.3.
Drag and drop the expressions into the boxes to correctly complete the proof of the polynomial identity.(x2 + y2)2 + 2x?y– y4 = x(x² + 4y?)(x2 + y2)2 + 2x²y2 – y4 = x(+ 4y?)+2x²y2 – y4 = x2 (x2 + 4y?)x² (x² + 47²)= x2 (x2 + 4y2)x² (x² + 47²) x² – 2x²y² + y x² + yt x² + 4x²72 x + 2x²,2x² + 2x²y² + yt
Answers
Answer:
x^4 + y^4 + 2x^2 y^2
x^4 + 4x^2y^2
x^2 (x^2 + 4y^2 )
Explanation:
Expanding the the expression gives
[tex]\begin{gathered} (x^2+y^2)^4=(x^2)^2+(y^2)^2+2(x^2)(y^2) \\ =\boxed{x^4+y^4+2x^2y^2\text{.}} \end{gathered}[/tex]Simplifying the Left-hand side gives
[tex]\begin{gathered} x^4+y^4+2x^2y^2+2x^2y^2-y^4 \\ =\boxed{x^4+4x^2y^2\text{.}} \end{gathered}[/tex]Finally, factoring out x^2 from the left-hand side gives
[tex]x^4+4x^2y^2=\boxed{x^2\mleft(x^2+4y^2\mright)\text{.}}[/tex]For which equation is x = 5 a solution ?
Answers
Given x = 5
We will find which equation will give a solution x = 5
1) x/2 = 10
so, x = 2 * 10 = 20
So, option (1) is wrong
2) x - 7 = 12
x = 12 + 7 = 19
So, option 2 is wrong
3) 2 + x = 3
x = 3 - 2 = 1
So, option 3 is wrong
4) 3x = 15
x = 15/3 = 5
So, the answer is option 3x = 15
Go step by step to reduce the radical. V 243 DVD try You must answer all questions above in order to submit.
Answers
We are given the following radical
[tex]\sqrt[]{243}[/tex]Let us reduce the above radical
We need to break the number 243 into a product of factors
Notice that 81 and 3 are the factors (83×3 = 243)
[tex]\sqrt[]{243}=\sqrt[]{81}\cdot\sqrt[]{3}[/tex]Since 81 is a perfect square so the radical becomes
[tex]\sqrt[]{243}=\sqrt[]{81}\cdot\sqrt[]{3}=9\cdot\sqrt[]{3}[/tex]Therefore, the simplified radical is
[tex]undefined[/tex]In general, the y-intercept of the function F(x) = a • bx is the point _____.A.(0, b)B.(0, a)C.(0, x)D.(0, 1)
Answers
The y-intercept of a function is the point where the function crosses the y axis and where x = 0
[tex]\begin{gathered} We\text{ are asked to find the y intercept of an exponential function, y = a*b}^x \\ When\text{ x = 0, b}^x\text{ =1 for any value of b} \\ We\text{ are then left with y = a*1 when x =0} \end{gathered}[/tex]The y intercept is therefore given by:
(0,a) --> option B
Solve for x: |x - 2| + 10 = 12 A x = 0 and x = 4B x = -4 and x = 0C x = -20 and x=4 D No solution
Answers
|x - 2| + 10 = 12
|x - 2| = 12 -10
|x - 2| = 2
There are 2 solutions:
x-2 = 2 and x-2 = -2
Solve each:
x = 2+2
x = 4
x-2=-2
x =-2+2
x=0
solution: x=0 and x = 4
Question 2 - Minimum Hours f. In the previous question, if Leah babysits for 7 hours this month, what is the minimum number of hours she would have to work at the ice cream shop to earn at least $120? Justify your answer. (4 POINTS) Give your answer to the nearest whole hour.
Answers
Leah earns 5x + 8y dollars, after x hours babysitting and y hours at the ice cream shop.
She wants to earn at least $120, then:
5x + 8y ≥ 120
Given that Leah babysits for 7 hours, then:
5(7) + 8y ≥ 120
35 + 8y ≥ 120
8y ≥ 120 - 35
8y ≥ 85
y ≥ 85/8
y ≥ 10.625
She must work at least 11 hours
Given that 7 + 11 = 18, then she would not work more than 20 hours as she expected
In the first week of July, a record 1,040 people went to the local swimming pool. In the second week,125 fewer people went to the pool than in the first week. In the third week,135 more people went to the pool than in the second week. In the fourth week,322 fewer people went to the pool than in the third week. What is the percent decrease in the number of people who went to the pool over these four weeks?
Answers
By the concept of percentage there is 20% decrease in the number of people who went to the pool over these four weeks.
What is percentage?A percentage is a statistic or ratio that is expressed as a fraction of 100 in mathematics. But even though the abbreviation "pct.", "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to signify it. A % is a dimensionless number; there is no specific unit of measurement for it. %, a relative figure signifying hundredths of any amount. Since one percent (symbolized as 1%) is equal to one hundredth of something, 100 percent stands for everything, and 200 percent refers to twice the amount specified. A percentage is a figure or ratio that in mathematics represents a portion of one hundred. It is frequently represented by the sign "%" or just "percent" or "pct." For instance, the fraction or decimal 0.35 is comparable to 35%.
In July:
First week:
Number of people went to the local swimming pool
=1040
Second week:
110 fewer people went to the pool than in the first week
Number of people went to the local swimming pool
=1040 - 110
=930
Third week:
130 more people went to the pool than in the second week
Number of people went to the local swimming pool
=930 + 130
=1060
Fourth week:
228 fewer people went to the pool than in the third week
Number of people went to the local swimming pool
=1060 - 228
=832
Decrease in number of people over four week
= number of people in first week - number of people in fourth week
Decrease in number of people over four week
=1040 - 832
=208
Now, the percentage
= 20%
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Let Iql = 5 at an angle of 45° and [r= 16 at an angle of 300°. What is 19-r|?13.00 14.2O 15.518.0
Answers
As given that:
[tex]|q|=5[/tex]At angle of45 degree
and |r| = 16 at 300 degree
so the |q| at 300 degree is:
[tex]\begin{gathered} |q|=5\times\frac{300}{45} \\ |q|=33.33 \end{gathered}[/tex]Now |q-r| is:
[tex]\begin{gathered} |q-r|=33.33-16 \\ |q-r|=17.33 \\ |q-r|\approx18 \end{gathered}[/tex]So the correct option is d.
The area of Square A is 36 square cm. The area of Square A’(A Prime) is 225 ᶜᵐ². What possible transformations did the square undergo?
Answers
A possible transformation is a scale. Since the area changed by
[tex]\frac{225}{36}=\frac{25}{4}[/tex]then a possible transformation was a scale by 25/4. A scale by a ratio bigger than one is a dilation.
Then the answer is B.
1.For a standard normal distribution, find:P(1.26 < z < 1.48)2.For a standard normal distribution, given:P(z < c) = 0.1288
Answers
Standard Normal Distribution
To find the cumulative probability of a Normal Distribution, we need to use some automated digital tool that makes the calculations for us, since it's a pretty complex formula.
We'll use an online tool and provide the results here.
a) P(1.26 < z < 1.48)
The procedure is: Find P(z < 1.48) directly from the tool. Find P(z < 1.48) also. Subtract both values.
P(z < 1.48) = 0.931
P(z < 1.26) = 0.896
Subtract the values above: 0.931 - 0.896 = 0.035. Thus:
P(1.26 < z < 1.48) = 0.035
b) Find c such that: P(z < c) = 0.1288
We need to use the inverse Normal Distribution, enter the probability and find the z-score: c = -1.132
Find the exact area of la circle if its circumference is 367 cm.
Answers
Given the circumference to be 367 cm.
Recall that the formula for the circumference of a circle is given as;
[tex]\begin{gathered} C=2\pi r \\ \Rightarrow367=2\pi r \end{gathered}[/tex]If we make r the subject of the formula,
[tex]r=\frac{367}{2\pi}[/tex]The area of a circle is given as;
[tex]\begin{gathered} A=\pi r^2 \\ \Rightarrow\pi(\frac{367}{2\pi})^2=\frac{\pi}{4\pi^2}(367)^2=10718.21 \end{gathered}[/tex]Does the point (–48, –47) satisfy the equation y = x − 1?
Answers
To find the answer to the question, we will substitute "-48" into "x" and "-47" into "y" and see if the equation holds true or not.
[tex]\begin{gathered} y=x-1 \\ -47\stackrel{?}{=}-48-1 \\ -47\neq-49 \end{gathered}[/tex]Thus, the point (-48, -47) does not satisfy the equation y = x - 1.
AnswerNoFind the roots of the equation 5x2 + 125 = 0
Answers
The given equation is:
[tex]5x^2+125=0[/tex]Divide through by 5
[tex]\begin{gathered} \frac{5x^2}{5}+\frac{125}{5}=\frac{0}{5} \\ \\ x^2+25=0 \\ \end{gathered}[/tex]This is further simplified as:
[tex]\begin{gathered} x^2=-25 \\ \\ \sqrt{x^2}=\pm\sqrt{-25} \\ \\ \sqrt{x^2}=\pm\sqrt{-1}\times\sqrt{25} \\ \\ x^=\pm5i \\ \\ x=5i\text{ and -5i} \\ \end{gathered}[/tex]The ratio between the radius of the base and the height of a cylinder is 2:3. If it's volume is 1617cm^3, find the total surface area of the cylinder.
Answers
Solution:
The ratio of the radius to the height of the cylinder is
[tex]2\colon3[/tex]Let the radius be
[tex]r=2x[/tex]Let the height be
[tex]h=3x[/tex]The volume of the cylinder is given below as
[tex]V=1617cm^3[/tex]Concept:
The volume of a cylinder is given below as
[tex]V_{\text{cylinder}}=\pi\times r^2\times h[/tex]By substituting values, we will have
[tex]\begin{gathered} V_{\text{cylinder}}=\pi\times r^2\times h \\ 1617=\frac{22}{7}\times(2x)^2\times(3x) \\ 1617=\frac{22}{7}\times4x^2\times3x \\ 1617\times7=264x^3 \\ \text{divdie both sides by 264} \\ \frac{264x^3}{264}=\frac{1617\times7}{264} \\ x^3=\frac{343}{8} \\ x=\sqrt[3]{\frac{343}{8}} \\ x=\frac{7}{2} \end{gathered}[/tex]The radius therefore will be
[tex]\begin{gathered} r=2x=2\times\frac{7}{2} \\ r=7cm \end{gathered}[/tex]The height of the cylinder will be
[tex]\begin{gathered} h=3x=3\times\frac{7}{2} \\ h=\frac{21}{2}cm \end{gathered}[/tex]The formula for the total surface area of a cylinder is given below as
[tex]T\mathrm{}S\mathrm{}A=2\pi r(r+h)[/tex]By substituting the values, we will have
[tex]\begin{gathered} TSA=2\pi r(r+h) \\ TSA=2\times\frac{22}{7}\times7(7+\frac{21}{2}) \\ TSA=44(7+\frac{21}{2}) \\ TSA=44\times7+44\times\frac{21}{2} \\ TSA=308+462 \\ TSA=770cm^2 \end{gathered}[/tex]Hence,
The total surface area of the cylinder is = 770cm²
write and slove six less then the product of a number n and 1/4 is no more than 96 fill in the boxs
Answers
ANSWER:
[tex]n\leq408[/tex]STEP-BY-STEP EXPLANATION:
With the statement we deduce the following inequality
[tex]\frac{1}{4}\cdot n-6\leq96[/tex]Solving for n
[tex]\begin{gathered} 4\cdot\frac{1}{4}\cdot n-4\cdot6\leq4\cdot96 \\ n-24\leq384 \\ n\leq384+24 \\ n\leq408 \end{gathered}[/tex]In 6-13 round each number to the place of the underlined digit
Answers
6. 32.7
7. 3.25
8. 41.1
9. 0.41
10. 6.1
11. 6.1
12. 184
13. 905.26
1) Considering that the underline marks the place to be rounded off we can do the following:
Note that if the number is greater than or equal to 5 then we will round it up.
If the number is lesser than 5 it will be rounded down.
Based on that we can round like this.
6. 32.7
7. 3.25
8. 41.1
9. 0.41
10. 6.1
11. 6.1
12. 184
13. 905.26
2)
Each expression below represents the area of a rectangle written as a product tha area model for each expression on your paper and label its length and then we nation showing that the area written as a product is equal to the area wimen as the the parts de prepared to share your equations with the class. a. (x+3)(2x + 1) ca(224) d. (2x + 5)(x + y + 2) • (20 - 1999 (2x-1) (2x-1) g. 2(325) ( 2+ y + 3) 2
Answers
We have the expression:
[tex]x(2x-y)[/tex]It can be thougth as the area of a rectangle with sides x and (2x-y).
We can also think of the the difference between an area of a rectangle with sides x and 2x and a rectangle with sides x and y:
[tex]x(2x-y)=x\cdot2x-x\cdot y=2x^2-xy[/tex]Answer: x(2x-y) = 2x^2-xy
The beam of light house makes one complete revolution every 20 seconds how many degrees is it rotate in five seconds
Answers
Answer:
Every 5 seconds the beam rotates 90 degrees;
[tex]90^{\circ}[/tex]Explanation:
Given that the beam of a lighthouse makes one complete revolution every 20 seconds.
one complete revolution is;
[tex]360^{\circ^{}}[/tex]The rate of rotation is;
[tex]r=\frac{360^{\circ}}{20}=18^{\circ}\text{ per second}[/tex]The number of degrees it will rotate in 5 seconds is;
[tex]\begin{gathered} x=5\times r=5\times18^{\circ}per\text{ second} \\ x=90^{\circ} \end{gathered}[/tex]Therefore, every 5 seconds the beam rotates 90 degrees;
[tex]90^{\circ}[/tex]laws of exponent : multiplication and power to a poweranswer and help me step by step
Answers
All the numbers multiply in the normal way, and the powers of a power need to be multiplied.
[tex]72c^4d^8e^{10}[/tex]what is 6q - q please
Answers
To solve this expression, we just have to subtract because they are like terms
[tex]6q-q=5q[/tex]Hence, the answer is 5q.