Question

1. surface areas of circular solids

the lateral area of a right cylinder having a base diameter of 10 units and a height of 10 units is
78.5 units².
157 units².
235.5 units².
314 units².

Explanation:

Step1: Recall lateral area formula

The lateral (curved) surface area of a right cylinder is given by \( LA = 2\pi rh \), where \( r \) is the radius and \( h \) is the height.
Given diameter \( d = 10 \) units, so radius \( r=\frac{d}{2}=\frac{10}{2}=5 \) units, and height \( h = 10 \) units.

Step2: Substitute values into formula

Substitute \( r = 5 \), \( h = 10 \), and \( \pi\approx3.14 \) into \( LA = 2\pi rh \):
\( LA = 2\times3.14\times5\times10 \)

Step3: Calculate the result

First, multiply \( 2\times5 = 10 \), then \( 10\times10 = 100 \), and finally \( 100\times3.14 = 314 \)? Wait, no—wait, \( 2\times3.14\times5\times10 = (2\times5)\times(3.14\times10)=10\times31.4 = 314 \)? Wait, no, wait: \( 2\times3.14\times5 = 31.4 \), then \( 31.4\times10 = 314 \)? Wait, no, that can't be. Wait, no—wait, maybe I made a mistake. Wait, lateral area of a cylinder is also equal to \( \pi dh \) (since \( d = 2r \), so \( 2\pi r h=\pi d h \)). Let's use \( \pi dh \): \( d = 10 \), \( h = 10 \), \( \pi\approx3.14 \). So \( LA=\pi\times10\times10 = 100\pi\approx314 \)? Wait, no, the options include 157. Wait, wait, maybe I messed up the formula. Wait, no—wait, lateral surface area of a cylinder: when you unroll the curved surface, it's a rectangle with height \( h \) and width equal to the circumference of the base (\( 2\pi r \) or \( \pi d \)). So the area is \( \text{width}\times\text{height}=\pi d\times h \). Wait, if \( d = 10 \), \( h = 10 \), then \( \pi\times10\times10 = 100\pi\approx314 \)? But the options have 157. Wait, wait, maybe the problem is using \( \pi = 3.14 \), but let's check again. Wait, radius is 5, height 10. \( 2\pi rh = 2\times3.14\times5\times10 = 314 \)? But the options include 157. Wait, maybe I misread the height? Wait, the problem says height of 10 units. Wait, maybe the formula is \( \pi r h \)? No, that's not correct. Wait, no—wait, lateral surface area of a cylinder is \( 2\pi rh \). Let's recalculate: \( 2\times3.14\times5\times10 \). \( 2\times5 = 10 \), \( 10\times10 = 100 \), \( 100\times3.14 = 314 \). But the options have 157. Wait, maybe the problem is using \( \pi = \frac{22}{7} \) or maybe I made a mistake. Wait, wait, 10 diameter, so radius 5. Height 10. \( 2\pi rh = 2\times3.14\times5\times10 = 314 \). But the options are 78.5, 157, 235.5, 314. Wait, 157 is half of 314. Wait, maybe the height is 5? No, the problem says height 10. Wait, no—wait, maybe the formula is \( \pi r^2 h \)? No, that's volume. Wait, lateral area is curved surface, not including the bases. Wait, maybe the question is about total surface area? No, the question says lateral area. Wait, let's check the options again. 78.5 is \( \pi r^2 \) (area of a circle), 157 is \( \pi r h \times 2 \)? Wait, no. Wait, \( 2\pi rh = 2\times3.14\times5\times10 = 314 \), but 157 is \( \pi\times5\times10\times2 / 2 \)? Wait, no. Wait, maybe the problem has a typo, or I misread. Wait, diameter 10, so radius 5. Height 10. \( 2\pi rh = 2\times3.14\times5\times10 = 314 \). So the correct option should be 314 units²? But let's check again. Wait, 3.14×5×10×2: 5×10=50, 50×2=100, 100×3.14=314. Yes.

Answer:

314 units² (the last option: 314 units²)