Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Sure, let's tackle the two parts of the question step by step.
4.1.3 Mpho's Claim:
Mpho claims that if the surface area of a sphere increases, then its radius will decrease. To evaluate this claim, we need to consider the formula for the surface area of a sphere.
The formula for the surface area ([tex]\(S\)[/tex]) of a sphere is given by:
[tex]\[ S = 4 \pi r^2 \][/tex]
where [tex]\(r\)[/tex] is the radius of the sphere.
Now, let's analyze the relationship:
1. If the surface area, [tex]\(S\)[/tex], increases, this implies that [tex]\(4 \pi r^2\)[/tex] must also increase.
2. To maintain the equation, [tex]\(r^2\)[/tex] needs to increase since [tex]\(4 \pi\)[/tex] is a constant.
3. Hence, for [tex]\(r^2\)[/tex] to increase, the radius [tex]\(r\)[/tex] itself must increase.
Therefore, if the surface area of a sphere increases, its radius will also increase. Consequently, Mpho's claim is incorrect.
4.2 Calculating the Common Interval of Ordinates:
Given the ordinates of an irregular metal plate in millimeters are:
[tex]\[ 120, 125, 135, 140, 145, 160, 155, 150, 145, 135, 118 \][/tex]
and the area of the plate is:
[tex]\[ 21135 \, \text{mm}^2 \][/tex]
We are to calculate the common interval of the ordinates.
Let's use the Trapezoidal Rule for an irregular shape. When applying the trapezoidal rule, the formula for the area ([tex]\(A\)[/tex]) with uniform intervals ([tex]\(h\)[/tex]) is given by:
[tex]\[ A = \frac{h}{2} \left[ y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n \right] \][/tex]
Where:
- [tex]\(y_0, y_1, \ldots, y_n\)[/tex] are the ordinates.
- [tex]\(h\)[/tex] is the common interval.
- [tex]\(n\)[/tex] is the number of intervals.
In this case, the sum of the ordinates is:
[tex]\[ y_0 + y_1 + y_2 + \cdots + y_{n-1} + y_n \][/tex]
Breaking it down:
- [tex]\(y_0 = 120\)[/tex]
- [tex]\(y_n = 118\)[/tex]
- sum of middle ordinates: [tex]\(125 + 135 + 140 + 145 + 160 + 155 + 150 + 145 + 135 = 1290\)[/tex]
- sum_ordinates including 2x middle ordinates: [tex]\(120 + 118 + 2 \times 1290\)[/tex]
Thus:
[tex]\[ y_0 + y_n + 2 \times \sum_{i=1}^{n-1} y_i = 120 + 118 + 2 \times 1290 = 2818 \][/tex]
Given the total area [tex]\(A = 21135 \, \text{mm}^2\)[/tex], we solve for [tex]\(h\)[/tex]:
[tex]\[ 21135 = \frac{h}{2} \times 2818 \][/tex]
[tex]\[ 21135 = 1409h \][/tex]
[tex]\[ h = \frac{21135}{1409} \][/tex]
[tex]\[ h = 15.0 \, \text{mm} \][/tex]
Given 11 ordinates creates 10 intervals, the number of intervals is indeed:
[tex]\[ 10 \][/tex]
Thus, the common interval [tex]\(h\)[/tex] is:
[tex]\[ 15.0 \, \text{mm} \][/tex]
In summary:
- Mpho's claim is incorrect because an increase in the surface area of a sphere will lead to an increase in its radius.
- The common interval of the ordinates given an area of [tex]\(21135 \, \text{mm}^2\)[/tex] is [tex]\(15.0 \, \text{mm} \)[/tex].
4.1.3 Mpho's Claim:
Mpho claims that if the surface area of a sphere increases, then its radius will decrease. To evaluate this claim, we need to consider the formula for the surface area of a sphere.
The formula for the surface area ([tex]\(S\)[/tex]) of a sphere is given by:
[tex]\[ S = 4 \pi r^2 \][/tex]
where [tex]\(r\)[/tex] is the radius of the sphere.
Now, let's analyze the relationship:
1. If the surface area, [tex]\(S\)[/tex], increases, this implies that [tex]\(4 \pi r^2\)[/tex] must also increase.
2. To maintain the equation, [tex]\(r^2\)[/tex] needs to increase since [tex]\(4 \pi\)[/tex] is a constant.
3. Hence, for [tex]\(r^2\)[/tex] to increase, the radius [tex]\(r\)[/tex] itself must increase.
Therefore, if the surface area of a sphere increases, its radius will also increase. Consequently, Mpho's claim is incorrect.
4.2 Calculating the Common Interval of Ordinates:
Given the ordinates of an irregular metal plate in millimeters are:
[tex]\[ 120, 125, 135, 140, 145, 160, 155, 150, 145, 135, 118 \][/tex]
and the area of the plate is:
[tex]\[ 21135 \, \text{mm}^2 \][/tex]
We are to calculate the common interval of the ordinates.
Let's use the Trapezoidal Rule for an irregular shape. When applying the trapezoidal rule, the formula for the area ([tex]\(A\)[/tex]) with uniform intervals ([tex]\(h\)[/tex]) is given by:
[tex]\[ A = \frac{h}{2} \left[ y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n \right] \][/tex]
Where:
- [tex]\(y_0, y_1, \ldots, y_n\)[/tex] are the ordinates.
- [tex]\(h\)[/tex] is the common interval.
- [tex]\(n\)[/tex] is the number of intervals.
In this case, the sum of the ordinates is:
[tex]\[ y_0 + y_1 + y_2 + \cdots + y_{n-1} + y_n \][/tex]
Breaking it down:
- [tex]\(y_0 = 120\)[/tex]
- [tex]\(y_n = 118\)[/tex]
- sum of middle ordinates: [tex]\(125 + 135 + 140 + 145 + 160 + 155 + 150 + 145 + 135 = 1290\)[/tex]
- sum_ordinates including 2x middle ordinates: [tex]\(120 + 118 + 2 \times 1290\)[/tex]
Thus:
[tex]\[ y_0 + y_n + 2 \times \sum_{i=1}^{n-1} y_i = 120 + 118 + 2 \times 1290 = 2818 \][/tex]
Given the total area [tex]\(A = 21135 \, \text{mm}^2\)[/tex], we solve for [tex]\(h\)[/tex]:
[tex]\[ 21135 = \frac{h}{2} \times 2818 \][/tex]
[tex]\[ 21135 = 1409h \][/tex]
[tex]\[ h = \frac{21135}{1409} \][/tex]
[tex]\[ h = 15.0 \, \text{mm} \][/tex]
Given 11 ordinates creates 10 intervals, the number of intervals is indeed:
[tex]\[ 10 \][/tex]
Thus, the common interval [tex]\(h\)[/tex] is:
[tex]\[ 15.0 \, \text{mm} \][/tex]
In summary:
- Mpho's claim is incorrect because an increase in the surface area of a sphere will lead to an increase in its radius.
- The common interval of the ordinates given an area of [tex]\(21135 \, \text{mm}^2\)[/tex] is [tex]\(15.0 \, \text{mm} \)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
