Question

plz help me????????? ​

Answer 1

AnswEr :

$\frak{Given}\begin{cases} \sf{\: Common \: difference \: (d) = 7}\\\sf{ \: 22nd \ term = 149} \end{cases}$

We've to find out the sum of 22 terms. So, n = 22

By using nth term Formula of the AP :

$\star \ \boxed{\sf{\purple{a_{n} = a + (n -1)d}}}$

$\underline{\bf{\dag} \:\mathfrak{Substituting \ Values \ in \ the \ formula \ :}}$

$:\implies\sf 149 = a + (22 - 1) \times 7 \\\\\\:\implies\sf 149 = a + 21 \times 7 \\\\\\:\implies\sf 149 = a + 147\\\\\\:\implies\sf a = 149 - 147\\\\\\:\implies\boxed{\frak{\purple{a = 2}}}$

$\therefore\underline{\textsf{ Here, we get value of the First term (a) of AP \textbf{2}}}.$

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For any Arithmetic Progression ( AP ), the sum of n terms is Given by :

$\bf{\dag}\quad\large\boxed{\sf S_n = \dfrac{n}{2}\bigg(a + l\bigg)}$

Where :

n = no. of terms

a = First Term

l = Last Term

$:\implies\sf S_{22} = \dfrac{\cancel{22}}{\cancel{\:2}} \bigg(2 + 149 \bigg) \\\\\\:\implies\sf S_{22} = 11 \times 151 \\\\\\:\implies\boxed{\frak{\purple{ S_{22} = 1661}}}$

$\therefore\underline{\textsf{ Hence, Sum of 22 terms of the AP is \textbf{1661}}}.$

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$\qquad\qquad{\underline{\underline{\dag \: \bf\:Formulaes \ of \: the \:AP\: :}}}$⠀⠀

To find out the nth term of the AP $\sf\pink{a_n + (n - 1)d}$⠀⠀⠀

⠀⠀

To find out the Sum of the AP = $\sf\purple{S_n = \dfrac{n}{2} \bigg[\sf 2a + (n - 1)d \bigg]}$⠀

To find out the sum of all terms have the last term of the AP 'l' = $\sf \blue{\dfrac{n}{2}(a + l)}.$

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Answer 2

Step-by-step explanation:

Figure refers to the attachment :-

\mathtt{\bf{\huge{\underline{\red{Question\:?}}}}}

Question?

★ The total surface area of hollow cylinder, which is open from both sides, is 4620 cm²; area of the base ring is 115.5 cm² and height is 7 cm. Find the thickness of the cylinder.

\mathtt{\bf{\huge{\underline{\green{Answer:-}}}}}

Answer:−

✒ The thickness of the cylinder is 0.368 cm.

\mathtt{\bf{\huge{\underline{\pink{Calcutaion:-}}}}}

Calcutaion:−

Given :-

The total surface area of hollow cylinder, which is open from both sides, is 4620 cm²

The area of the base ring is 115.5 cm².

The height is 7 cm.

To Find :-

The thickness of the cylinder.

Solution :-

Let the radius of outer surface be R

& the radius of inner surface are r .

∴ Area of the base ring = π(R² - r²)

➝ 115.5 = π(R² - r²)

➝ (R² - r²) = 115.5 ÷ 22/7

➝ (R² - r²) = \dfrac{1155 × 7}{22}

22

1155×7

➝(R + r) (R - r) = \dfrac{1155 × 7}{220}

220

1155×7

(R + r) (R - r) = \dfrac{147}{4}

4

147

cm²______{1}

According to the question,

Total surface area of the cylinder = 4620 sq cm

★ We know that the total surface area of a hollow cylinder = (outer curved surface of cylinder + inner curved surface area of cylinder ) + 2( The circular base area of cylinder )

➝ 2πRh + 2πrh + 2π(R² - r²)

➝ 2πRh + 2πrh + 2π(R² - r²) = 4620

➝ 2πh (R + r) + (2 × 115.5) = 4620

➝ 2πh (R + r) + 231 = 4620

➝ 2πh (R + r) = 4620 - 231

➝ 2 × 22/7 × 7 × (R + r) = 4389

➝ (R + r) = \dfrac{4389}{44}

44

4389

➝ (R + r) = \dfrac{399}{4}

4

399

__[2]

Putting value of 2 in equation (1)

➝ (R + r)(R - r) = \dfrac{147}{4}

4

147

➝ \dfrac{399}{4}

4

399

(R - r) = \dfrac{147}{4}

4

147

➝(R - r) = \dfrac{147}{4}

4

147

÷ \dfrac{399}{4}

4

399

➝ (R - r) = \dfrac{147}{4}

4

147

× \dfrac{4}{399}

399

4

➝ (R - r) = \dfrac{147}{399}

399

147

➝ (R - r) = \dfrac{7}{19}

19

7

cm

➝ (R - r) = 0.368 cm

Therefore, the cylinder's thickness is 0.368 cm.

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Information Regarding Question :-

We have a cylinder with its dimensions, we have to find its thickness , by applying formulae of cylinder we have to find.

Area of the base ring = π(R² - r²)

The total surface area of a hollow cylinder = (outer curved surface of cylinder + inner curved surface area of cylinder ) + 2( The circular base area of cylinder )

Application :-

In physics practical .

Making sports goods.

Used in Industries .

Making bangles.

Making coils.

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