Question

Here is a prism abcdefgh. The base abcd of the prism is a square of side 18cm. x is the point on cd such that dx : xc = 5 : 4. The cross section of the prism is in the shape of a trapezium of area 203cm². gh = 11cm. Find the size of the angle between the line gx and the base abcd. Give your answer to 1 decimal place.

Answer 1

To find the size of the angle between the line gx and the base abcd in the given prism, we first determine the lengths of dx and xc. Then, we calculate the area of the trapezium cross section. Finally, we use the tangent ratio to find the size of the angle. we find that the size of the angle between gx and the base abcd is approximately 52.7 degrees.

Step-by-step explanation:

To find the size of the angle between the line gx and the base abcd, we need to use geometric properties and relationships. Let's start by finding the lengths of dx and xc. Since dx : xc = 5 : 4 and dc = 18 cm, we can calculate the length of dx by dividing 5/9 of 18 cm. This gives us dx = 10 cm, and consequently, xc = 8 cm.

Next, we need to find the area of the trapezium cross section. Since the area is given as 203 cm², we can use the formula for the area of a trapezium: A = 1/2 × (sum of parallel sides) × height. Plugging in the given values, we get 203 = 1/2 × (18 + gh) × 18. Solving for gh, we find gh = 11 cm.

Now we have all the information we need. To find the angle between gx and the base abcd, we can use the tangent ratio. Tangent(angle) = opposite/adjacent. In this case, the opposite side is gh (11 cm) and the adjacent side is xc (8 cm). Plug the values into the equation: tan(angle) = 11/8. Solving for the angle, we find that the size of the angle between gx and the base abcd is approximately 52.7 degrees.

Answer 2

To find the size of the angle between the line gx and the base abcd, we can use the fact that the cross section of the prism is in the shape of a trapezium of area 203cm².

Step-by-step explanation:

To find the size of the angle between the line gx and the base abcd, we can use the fact that the cross section of the prism is in the shape of a trapezium of area 203cm². The area of a trapezium is given by the formula A = 1/2(a + b)h, where a and b are the lengths of the parallel sides and h is the distance between them. In this case, we can let a = 18cm (side length of the base abcd) and h = 11cm (height of prism gh). We need to find the length of side b.

We know that dx : xc = 5 : 4, so we can calculate dx as follows:

dx = 5/(5+4) * 18cm = 9cm

Using the formula for the area of a trapezium, we can now solve for b:

203cm² = 1/2(18cm + b)(11cm)

406cm² = (18cm + b)(11cm)

406cm² = 198cm + 11b

208cm² = 11b

b = 208/11 cm ≈ 18.909 cm

Now that we have the lengths of all sides of the trapezium, we can use trigonometry to find the angle between gx and the base abcd. Let's call this angle α.

sin(α) = h/b = 11/18.909

α = arcsin(11/18.909) ≈ 38.1 degrees