Calculate for me this:1.An open water tank is in shape of a frustum of a square -based pyramid.the bottom and the top are 6m and 8m long respectively.the tank has a depth of 2m.Determine the total surface area of the tank. 2.A kitchen workshop is in shape of a frustum of a rectangular-based pyramid.The bottom measurements are 100cm and 150cm and the top measures 60cm by 90cm.The height of the kitchen workshop is 200cm. Determine the total surface area of the kitchen workshop. 3.joy made a hollow lampshade in the shape of a square -based frustum of a pyramid the top and the bottom sides of the lampshade measured 10cm and 18cm respectively.The height of the lampshade was 16cm, determine the area of material required to make half a dozen similar lampshade.

Eneo la msingi wa chini ni mraba.

$$\text{Eneo la msingi} = s_1^2 = (6 \text{ m})^2 = 36 \text{ m}^2$$

Tunatumia theorem ya Pythagoras. Tofauti ya nusu-urefu wa pande ni $\frac{s_2 - s_1}{2}$.

$$\text{Tofauti ya nusu-urefu} = \frac{8 \text{ m} - 6 \text{ m}}{2} = 1 \text{ m}$$ $$l = \sqrt{h^2 + \left(\frac{s_2 - s_1}{2}\right)^2} = \sqrt{(2 \text{ m})^2 + (1 \text{ m})^2} = \sqrt{4 \text{ m}^2 + 1 \text{ m}^2} = \sqrt{5} \text{ m}$$

Kuna nyuso nne za pembeni.

$$\text{Jumla ya eneo la pembeni} = 4 \times 7\sqrt{5} \text{ m}^2 = 28\sqrt{5} \text{ m}^2 \approx 28 \times 2.236067977 \text{ m}^2 \approx 62.6099 \text{ m}^2$$

Hii ni jumla ya eneo la msingi wa chini na jumla ya eneo la nyuso za pembeni.

$$\text{Jumla ya eneo} = 36 \text{ m}^2 + 28\sqrt{5} \text{ m}^2 \approx 36 \text{ m}^2 + 62.6099 \text{ m}^2 = 98.6099 \text{ m}^2$$

Tunazungusha hadi sehemu mbili za desimali.

$$\text{Jumla ya eneo} \approx 98.61 \text{ m}^2$$

Jumla ya eneo la uso wa tanki ni $\boxed{\text{98.61 m}^2}$.

2. Warsha ya Jikoni (Frustum ya Piramidi yenye Msingi wa Mstatili)

Data:

• Vipimo vya chini: urefu ($l_1$) = 150 cm, upana ($w_1$) = 100 cm
• Vipimo vya juu: urefu ($l_2$) = 90 cm, upana ($w_2$) = 60 cm
• Urefu ($h$): 200 cm
• Hii ni frustum iliyofungwa, kwa hivyo tunahitaji eneo la msingi wa chini, msingi wa juu, na nyuso nne za pembeni.

$$\text{Eneo la msingi wa chini} = l_1 \times w_1 = 150 \text{ cm} \times 100 \text{ cm} = 15000 \text{ cm}^2$$ $$\text{Eneo la msingi wa juu} = l_2 \times w_2 = 90 \text{ cm} \times 60 \text{ cm} = 5400 \text{ cm}^2$$

Hizi ni nyuso zenye upana $w_1, w_2$. Tofauti ya nusu-upana ni $\frac{w_1 - w_2}{2}$.

$$\text{Tofauti ya nusu-upana} = \frac{100 \text{ cm} - 60 \text{ cm}}{2} = 20 \text{ cm}$$ $$l_w = \sqrt{h^2 + \left(\frac{w_1 - w_2}{2}\right)^2} = \sqrt{(200 \text{ cm})^2 + (20 \text{ cm})^2} = \sqrt{40000 + 400} = \sqrt{40400} \text{ cm}$$

Kila uso ni trapezoid.

$$\text{Jumla ya eneo la nyuso mbili (urefu)} = 2 \times \frac{1}{2}(l_1 + l_2)l_w = (150 \text{ cm} + 90 \text{ cm})\sqrt{40400} \text{ cm}$$ $$= 240\sqrt{40400} \text{ cm}^2 \approx 240 \times 200.9975124 \text{ cm}^2 \approx 48239.40 \text{ cm}^2$$

Hizi ni nyuso zenye urefu $l_1, l_2$. Tofauti ya nusu-urefu ni $\frac{l_1 - l_2}{2}$.

$$\text{Tofauti ya nusu-urefu} = \frac{150 \text{ cm} - 90 \text{ cm}}{2} = 30 \text{ cm}$$ $$l_l = \sqrt{h^2 + \left(\frac{l_1 - l_2}{2}\right)^2} = \sqrt{(200 \text{ cm})^2 + (30 \text{ cm})^2} = \sqrt{40000 + 900} = \sqrt{40900} \text{ cm}$$

Kila uso ni trapezoid.

$$\text{Jumla ya eneo la nyuso mbili (upana)} = 2 \times \frac{1}{2}(w_1 + w_2)l_l = (100 \text{ cm} + 60 \text{ cm})\sqrt{40900} \text{ cm}$$ $$= 160\sqrt{40900} \text{ cm}^2 \approx 160 \times 202.2374846 \text{ cm}^2 \approx 32357.99 \text{ cm}^2$$

$$\text{Jumla ya eneo} = \text{Eneo la msingi wa chini} + \text{Eneo la msingi wa juu} + \text{Jumla ya eneo la nyuso mbili (urefu)} + \text{Jumla ya eneo la nyuso mbili (upana)}$$ $$\text{Jumla ya eneo} = 15000 \text{ cm}^2 + 5400 \text{ cm}^2 + 240\sqrt{40400} \text{ cm}^2 + 160\sqrt{40900} \text{ cm}^2$$ $$\text{Jumla ya eneo} \approx 15000 + 5400 + 48239.40 + 32357.99 \text{ cm}^2 = 100997.39 \text{ cm}^2$$

Tunazungusha hadi sehemu mbili za desimali.

$$\text{Jumla ya eneo} \approx 100997.40 \text{ cm}^2$$

Jumla ya eneo la uso wa warsha ya jikoni ni $\boxed{\text{100997.40 cm}^2}$.

3. Lampshade ya Joy (Frustum ya Piramidi yenye Msingi wa Mraba)

Data:

• Urefu wa upande wa juu ($s_2$): 10 cm
• Urefu wa upande wa chini ($s_1$): 18 cm
• Urefu ($h$): 16 cm
• Lampshade ni tupu (hollow), kwa hivyo tunahitaji tu eneo la nyuso za pembeni.
• Idadi: Nusu dazeni = 6 lampshades.

Tofauti ya nusu-urefu wa pande ni $\frac{s_1 - s_2}{2}$.

$$\text{Tofauti ya nusu-urefu} = \frac{18 \text{ cm} - 10 \text{ cm}}{2} = 4 \text{ cm}$$ $$l = \sqrt{h^2 + \left(\frac{s_1 - s_2}{2}\right)^2} = \sqrt{(16 \text{ cm})^2 + (4 \text{ cm})^2} = \sqrt{256 \text{ cm}^2 + 16 \text{ cm}^2} = \sqrt{272} \text{ cm}$$

Kuna nyuso nne za pembeni.

$$\text{Jumla ya eneo la pembeni (lampshade moja)} = 4 \times \frac{1}{2}(s_1 + s_2)l = 2(18 \text{ cm} + 10 \text{ cm})\sqrt{272} \text{ cm}$$ $$= 2(28)\sqrt{272} \text{ cm}^2 = 56\sqrt{272} \text{ cm}^2 \approx 56 \times 16.4924225 \text{ cm}^2 \approx 923.5757 \text{ cm}^2$$

$$\text{Jumla ya eneo la nyenzo} = 6 \times 56\sqrt{272} \text{ cm}^2 = 336\sqrt{272} \text{ cm}^2$$ $$\text{Jumla ya eneo la nyenzo} \approx 336 \times 16.4924225 \text{ cm}^2 \approx 5541.45396 \text{ cm}^2$$

Tunazungusha hadi sehemu mbili za desimali.

$$\text{Jumla ya eneo la nyenzo} \approx 5541.45 \text{ cm}^2$$

Jumla ya eneo la nyenzo zinazohitajika ni $\boxed{\text{5541.45 cm}^2}$.