1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
kirza4 [7]
4 years ago
12

A)A cuboid with a square x cm and height 2xcm². Given total surface area of the cuboid is 129.6cm² and x increased at 0.01cms-¹.

Find the change of the volume the cuboid
b) Given y=2x²+3x, use differentiation to find small change in y when x increased from 4 to 4.02.
Mathematics
1 answer:
Nutka1998 [239]4 years ago
6 0

Answer: (given assumed typo corrections)


(V ∘ X)'(t) = 0.06(0.01t+3.6)^2 cm^3/sec.


The rate of change of the volume of the cuboid in change of volume per change in seconds, after t seconds. Not a constant, for good reason.



Part B) y'(x+Δx/2)×Δx gives exactly the same as y(x+Δx)-y(x), 0.3808, since y is quadratic in x so y' is linear in x.


Step-by-step explanation:

This problem has typos. Assuming:

Cuboid has square [base with side] X cm and height 2X cm [not cm^2]. Total surface area of cuboid is 129.6 cm^2, and X [is] increas[ing] at rate 0.01 cm/sec.


129.6 cm^2 = 2(base cm^2) + 4(side cm^2)

= 2(X cm)^2 + 4(X cm)(2X cm)

= (2X^2 + 8X^2)cm^2

= 10X^2 cm^2

X^2 cm^2 = 129.6/10 = 12.96 cm^2

X cm = √12.96 cm = 3.6 cm


so X(t) = (0.01cm/sec)(t sec) + 3.6 cm, or, omitting units,

X(t) = 0.01t + 3.6

= the length parameter after t seconds, in cm.


V(X) = 2X^3 cm^3

= the volume when the length parameter is X.


dV(X(t))/dt = (dV(X)/dX)(X(t)) × dX(t)/dt

that is, (V ∘ X)'(t) = V'(X(t)) × X'(t) chain rule


V'(X) = 6X^2 cm^3/cm

= the rate of change of volume per change in length parameter when the length parameter is X, units cm^3/cm. Not a constant (why?).


X'(t) = 0.01 cm/sec

= the rate of change of length parameter per change in time parameter, after t seconds, units cm/sec.

V(X(t)) = (V ∘ X)(t) = 2(0.01t+3.6)^3 cm^3

= the volume after t seconds, in cm^3

V'(X(t)) = 6(0.01t+3.6)^2 cm^2

= the rate of change of volume per change in length parameter, after t seconds, in units cm^3/cm.

(V ∘ X)'(t) = ( 6(0.01t+3.6)^2 cm^3/cm )(0.01 cm/sec) = 0.06(0.01t+3.6)^2 cm^3/sec

= the rate of change of the volume per change in time, in cm^3/sec, after t seconds.


Problem to ponder: why is (V ∘ X)'(t) not a constant? Does the change in volume of a cube per change in side length depend on the side length?


Question part b)


Given y=2x²+3x, use differentiation to find small change in y when x increased from 4 to 4.02.


This is a little ambiguous, but "use differentiation" suggests that we want y'(4.02) yunit per xunit, rather than Δy/Δx = (y(4.02)-y(4))/(0.02).


Neither of those make much sense, so I think we are to estimate Δy given x and Δx, without evaluating y(x) at all.

Then we want y'(x+Δx/2)×Δx


y(x) = 2x^2 + 3x

y'(x) = 4x + 3


y(4) = 44

y(4.02) = 44.3808

Δy = 0.3808

Δy/Δx = (0.3808)/(0.02) = 19.04


y'(4) = 19

y'(4.01) = 19.04

y'(4.02) = 19.08


Estimate Δy = (y(x+Δx)-y(x)/Δx without evaluating y() at all, using only y'(x), given x = 4, Δx = 0.02.


y'(x+Δx/2)×Δx = y'(4.01)×0.02 = 19.04×0.02 = 0.3808.


In this case, where y is quadratic in x, this method gives Δy exactly.

You might be interested in
Line L passes through the point (-1, 1) and is PARALLEL to the line shown here. An equation of line L is
Pachacha [2.7K]
The problem cannot be fully solved because you did provide the line on which the line is parallel/ but i can give you the steps on how to solve it. first solve the slope of the parallel line by
m = ( y2 - y1 ) / ( x2 - x1 )
then the slope of that line is equal to slope of line passing pooint ( -1 , 1)
then solve the y intercept of the line using
y = mx + b
where b is the y intercept
then you will have the equation of the line
5 0
3 years ago
Read 2 more answers
Michael just drank a cup of coffee to help him stay awake. The coffee had 110 milligrams of caffeine in it. If his body processe
GarryVolchara [31]

the answer is 74 goodluck



3 0
3 years ago
Read 2 more answers
Sam opened a money-market account that pays 3% simple interest. He started the account with $7,000 and made no further deposits.
Verizon [17]
He hold account for 5 years or 5 months. Depending on 3% per year or 3% per month
6 0
3 years ago
In the equation (x^2+y)^5, what is the coefficient of the term x^4y^3? what is the coefficient of the same term in the expansion
lys-0071 [83]

\displaystyle
(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^{n-k}y^k

<em>-------------------------------------------------------------</em>


\displaystyle&#10;(x^2+y)^n=\sum_{k=0}^n\binom{n}{k}x^{2n-2k}y^k\\&#10;n=5\\&#10;k=3\\\\\binom{5}{3}=\dfrac{5!}{3!2!}=\dfrac{4\cdor5}{2}=10

<u>It's 10.</u>

----------------------------------------------------

\displaystyle&#10;(3x^2+y)^n=\sum_{k=0}^n\binom{n}{k}(3x)^{2n-2k}y^k=\sum_{k=0}^n\binom{n}{k}\cdot 3^{2n-2k}\cdot x^{2n-2k}y^k\\\\&#10;n=5\\&#10;k=4\\\\&#10;\binom{5}{3}\cdot3^{2\cdot5-2\cdot4}=10\cdot3^{2}=10\cdot9=90

<u>It's 90</u>

6 0
3 years ago
Subtract the given function and indicate the domain of the difference
maxonik [38]

Answer:

The domain is x\in (-\infty,\infty).

Step-by-step explanation:

Given functions f(x)=x^2+3x+1 and g(x)=2x^2-4x-1

Subtract these two functions:

f(x)-g(x)\\ \\=(x^2+3x+1)-(2x^2-4x-1)\\ \\=x^2+3x+1-2x^2+4x+1\\ \\=(x^2-2x^2)+(3x+4x)+(1+1)\\ \\=-x^2+7x+2

Plot these difference on the coordinate plane (see attached diagram). This function is defined for all vlues of x, so the domain is x\in (-\infty,\infty).

6 0
3 years ago
Other questions:
  • The sum of two consecutive odd integers equals 12. What are they?
    15·1 answer
  • Given the equation of the line (y-2)=3(x+1) what is the slope
    11·1 answer
  • The quotient of twice a number t and 12
    14·1 answer
  • Can someone please help me on this ????
    14·1 answer
  • The ordered pairs (0, 2) and (1,6)
    12·1 answer
  • Please make sure you do it right.
    14·2 answers
  • Hello! its me again!!!
    5·1 answer
  • I WILL GIVE BRAINLIEST. PLEASE HELP RIGHT NOW
    10·1 answer
  • Select the two statements that are true about the equation y+6=−10(x−3).
    14·1 answer
  • suppose that the function f(x) = 5.32 0.80x represents the cost of mailing an object that weighs x pounds. how much would it cos
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!