1. find the lateral area of a right prism whose altitude measures 20 cm and whose base is a square with a width 7 cm long.
2. the volume of a rectangular solid is 5376 cubic meters, and the base is 24 meters by 16 meters. find the height of the solid.
3. a rectangle has a perimeter of 22cm and an area of 30 cubic meter. what are its dimensions?
1. The lateral area is the sum of the
areas of the vertical sides.
Al = 2(W*h) + 2(L*h).
Al = 2( 7*20) + 2(7*20) = 560 cm^2.
2. V = L*W*h.
22*16*h = 5376.
Solve for h.
3. 2W + 2L = 22,
Divide both sides by 2:
L + W = 11,
L = 11-W.
L*W = 30,
Substitute 11-W for L:
(11-W)W = 30,
-W^2+11W-30 = 0,
W^2-11W+30 = 0,
(W-5)(w-6) = 0,
W = 5 or 6.
L+5 = 11,
L = 6.
Dimensions = 6 x 5.
1. Ah, the lateral area of a right prism, a classic! But square base, huh? I guess it's trying to be unique. Well, to find the lateral area, we need to find the perimeter of the base first. Since it's a square, all sides are equal (7 cm). So, the perimeter is just 4 times one side: 4 x 7 = 28 cm.
Now, we multiply the perimeter by the height (altitude): 28 cm x 20 cm = 560 cm². And there you have it, the lateral area of our prism with a square base!
2. Ah, the volume of a rectangular solid, an old friend. So, to find the height, we need to divide the volume by the base area. Given that the base is 24 meters by 16 meters, the area is 24 m x 16 m = 384 square meters.
Now, we divide the volume (5376 cubic meters) by the base area (384 square meters): 5376 m³ ÷ 384 m² = 14 meters. So, the height of our rectangular solid stands tall at 14 meters!
3. Ah, the mysterious rectangle with a secret perimeter and area. To solve this riddle, let's dive in! Now, we know that the perimeter is 22 cm, which means the sum of all sides is 22 cm. Since opposite sides of a rectangle are equal, we can say that 2 times the length plus 2 times the width equals 22 cm.
Let's call the length L and the width W. So, we have 2L + 2W = 22. Now, let's move on to the area, which is 30 square meters. The area of a rectangle is the product of its length and width, so L times W = 30.
Now, let's solve this humorously mathematical mystery! We have a system of equations:
2L + 2W = 22
LW = 30
Hmm, let me put on my clown hat and wiggle my nose... abracadabra! The dimensions of the rectangle are L = 5 cm and W = 6 cm! Ta-da!
1. To find the lateral area of a right prism, you need to calculate the perimeter of the base and multiply it by the height of the prism.
Given:
Altitude (height) of the prism = 20 cm
Width of the square base = 7 cm
First, find the perimeter of the base:
Perimeter of a square = 4 * side length
Perimeter of the base = 4 * 7 cm = 28 cm
Now, calculate the lateral area:
Lateral Area = Perimeter of the base * Height
Lateral Area = 28 cm * 20 cm = 560 cm²
Therefore, the lateral area of the right prism is 560 cm².
2. To find the height of a rectangular solid, divide the volume of the solid by the product of the length and width of the base.
Given:
Volume of the solid = 5376 cubic meters
Length of the base = 24 meters
Width of the base = 16 meters
Now, calculate the height:
Height = Volume / (Length * Width)
Height = 5376 m³ / (24 m * 16 m)
Height = 5376 m³ / 384 m²
Height = 14 meters
Therefore, the height of the rectangular solid is 14 meters.
3. To find the dimensions of a rectangle with a given perimeter and area, you need to set up and solve a system of equations.
Given:
Perimeter of the rectangle = 22 cm
Area of the rectangle = 30 cm²
Let's assume the length of the rectangle is L and the width is W.
From the given information, we can set up two equations:
Equation 1: 2L + 2W = 22 (perimeter equation)
Equation 2: L * W = 30 (area equation)
Solve Equation 1 for L:
2L + 2W = 22
2L = 22 - 2W
L = (22 - 2W) / 2
L = 11 - W
Substitute this value of L in Equation 2:
(11 - W) * W = 30
11W - W² = 30
Rearrange Equation 3 to get a quadratic equation:
W² - 11W + 30 = 0
Now, we can solve this equation using factoring or the quadratic formula:
(W - 5)(W - 6) = 0
This gives two possible values for W:
W = 5 or W = 6
Substitute these values back into Equation 1 to find the respective lengths:
For W = 5, L = 11 - 5 = 6
For W = 6, L = 11 - 6 = 5
Therefore, the dimensions of the rectangle could be either 6 cm by 5 cm or 5 cm by 6 cm.
1. To find the lateral area of a right prism, you need to calculate the perimeter of the base and multiply it by the height. Let's break it down step by step for your specific problem:
- The altitude of the prism is given as 20 cm.
- The base is a square with a width of 7 cm. Since it's a square, all the sides are equal.
- The perimeter of a square is found by multiplying the width by 4. In this case, the perimeter is 7 cm * 4 = 28 cm.
- Finally, to find the lateral area, you multiply the perimeter of the base by the height: 28 cm * 20 cm = 560 cm².
Therefore, the lateral area of the prism is 560 cm².
2. To find the height of a rectangular solid given its volume and base dimensions, you need to divide the volume by the product of the length and width of the base. Here is how you can solve it:
- The volume of the solid is given as 5376 cubic meters.
- The base dimensions are given as 24 meters by 16 meters.
- The formula for volume of a rectangular solid is length * width * height.
- Rearranging the formula, we can solve for height: height = volume / (length * width).
- Plug in the values: height = 5376 m³ / (24 m * 16 m).
- Simplify: height = 5376 m³ / 384 m².
- Divide: height = 14 m.
Therefore, the height of the rectangular solid is 14 meters.
3. To find the dimensions of a rectangle given its perimeter and area, you need to set up a system of equations using the given information. Here are the steps to solve it:
- The perimeter of the rectangle is given as 22 cm.
- The area of the rectangle is given as 30 square meters.
- The formula for the perimeter of a rectangle is 2 * (length + width).
- In this case, for simplicity, let's consider length as L and width as W.
- Set up the equation for the perimeter: 2 * (L + W) = 22.
- Simplify: L + W = 11.
- The formula for the area of a rectangle is length * width.
- Set up the equation for the area: L * W = 30.
- Use substitution to solve the system of equations. Solve one equation for L or W, and substitute it into the other equation.
- For example, solve the first equation for W: W = 11 - L.
- Substitute the value of W into the second equation: L * (11 - L) = 30.
- Rearrange and solve the quadratic equation: L² - 11L + 30 = 0.
- Factorize the quadratic equation: (L - 5)(L - 6) = 0.
- Solve for L: L = 5 or L = 6.
- Substitute the values of L back into the first equation to find W: W = 11 - L.
- For L = 5, W = 11 - 5 = 6.
- For L = 6, W = 11 - 6 = 5.
Therefore, the dimensions of the rectangle are 5 cm by 6 cm.