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WARRIOR [948]
2 years ago
9

If X = 2 inches, Y = 8 inches, W = 4 inches, and Z = 4 inches, what is the area of the object?

Mathematics
1 answer:
max2010maxim [7]2 years ago
4 0

Answer:

C. 63 square inches

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Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n.
Otrada [13]

I guess the "5" is supposed to represent the integral sign?

I=\displaystyle\int_1^4\ln t\,\mathrm dt

With n=10 subintervals, we split up the domain of integration as

[1, 13/10], [13/10, 8/5], [8/5, 19/10], ... , [37/10, 4]

For each rule, it will help to have a sequence that determines the end points of each subinterval. This is easily, since they form arithmetic sequences. Left endpoints are generated according to

\ell_i=1+\dfrac{3(i-1)}{10}

and right endpoints are given by

r_i=1+\dfrac{3i}{10}

where 1\le i\le10.

a. For the trapezoidal rule, we approximate the area under the curve over each subinterval with the area of a trapezoid with "height" equal to the length of each subinterval, \dfrac{4-1}{10}=\dfrac3{10}, and "bases" equal to the values of \ln t at both endpoints of each subinterval. The area of the trapezoid over the i-th subinterval is

\dfrac{\ln\ell_i+\ln r_i}2\dfrac3{10}=\dfrac3{20}\ln(ell_ir_i)

Then the integral is approximately

I\approx\displaystyle\sum_{i=1}^{10}\frac3{20}\ln(\ell_ir_i)\approx\boxed{2.540}

b. For the midpoint rule, we take the rectangle over each subinterval with base length equal to the length of each subinterval and height equal to the value of \ln t at the average of the subinterval's endpoints, \dfrac{\ell_i+r_i}2. The area of the rectangle over the i-th subinterval is then

\ln\left(\dfrac{\ell_i+r_i}2\right)\dfrac3{10}

so the integral is approximately

I\approx\displaystyle\sum_{i=1}^{10}\frac3{10}\ln\left(\dfrac{\ell_i+r_i}2\right)\approx\boxed{2.548}

c. For Simpson's rule, we find a quadratic interpolation of \ln t over each subinterval given by

P(t_i)=\ln\ell_i\dfrac{(t-m_i)(t-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+\ln m_i\dfrac{(t-\ell_i)(t-r_i)}{(m_i-\ell_i)(m_i-r_i)}+\ln r_i\dfrac{(t-\ell_i)(t-m_i)}{(r_i-\ell_i)(r_i-m_i)}

where m_i is the midpoint of the i-th subinterval,

m_i=\dfrac{\ell_i+r_i}2

Then the integral I is equal to the sum of the integrals of each interpolation over the corresponding i-th subinterval.

I\approx\displaystyle\sum_{i=1}^{10}\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt

It's easy to show that

\displaystyle\int_{\ell_i}^{r_i}P(t_i)\,\mathrm dt=\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)

so that the value of the overall integral is approximately

I\approx\displaystyle\sum_{i=1}^{10}\frac{r_i-\ell_i}6(\ln\ell_i+4\ln m_i+\ln r_i)\approx\boxed{2.545}

4 0
3 years ago
SOMEONE PLEASEEEEE HELPPPPP MEEEEE
Hatshy [7]

Answer:

(5,3)

There is no explanation for this solution.

8 0
3 years ago
Read 2 more answers
PLEASE HELP ME PLZ!!​
vichka [17]

Answer:

1.  Formula is A2 : A9 = COUNT( A2: A9 ) = 8

2. Formula is SUM( A2: A9 ) = 36

3. Formula is B2 : B9 = COUNT( B2: B9) = 8

4. Formula is  MAX( C2: C9) = 5

5. Formula is MIN( C4: C8) = 3

6. Formula is SUM( C5 - C6) = 0

7. Formula is AVERAGE( C2: C9) = 4

Step-by-step explanation: Have a nice day! ✌️

7 0
2 years ago
This system of equations has A. exactly one solution B. no solution C. infinitely many solution​
german

Answer:

C.

Step-by-step explanation:

This is because both lines are parallel, leaving infinite solutions.

5 0
3 years ago
P = 5(2t -20) what is an equivalent form of this rule
puteri [66]

Answer:

p = 10t - 100

Step-by-step explanation:

Perform the indicated multiplication:  p = 10t - 100.

3 0
2 years ago
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