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CAHSEE practice: Problems 35-37

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Problem 35. In the figure below, every angle is a right angle. So they're just saying that's a right angle, that's a right angle, these are all right angles. You get the idea. What is the area, in square units, of the figure? So the area. How can we figure this out? We can divide this up into a bunch of rectangles and figure out their respective areas. So if we drew a rectangle right here, that rectangle, what's its area? It's going to be 6 times 8, which is 48. That's that rectangle right there, is 48 square units. Let's see what else we can do. We can do this rectangle right here. That rectangle right there is also 6 by 8, so this is also going to be 48 square units right there. And then we have-- we'll do it in another color. We have this rectangle-- that rectangle right there is divided right there-- we have this rectangle right here, which is 3 by 4. So 3 times 4 is 12 units. And then you have this rectangle right here, which is 3 by 4. So it's also 12 square units. So the whole area of the entire thing is 12 plus 48, which is 60. Right? 2 plus 48 is 50, right, that's 60. So you have 60 there, and then you have another 48, gives you 108, plus 12, which is 120. So the area in square units of the figure is 120. Problem 36. A rectangular field is 363 feet long and 240 feet wide. How many acres is the field? So let me draw this field. I'll do it in green because we're talking about a field. So it is 363 feet long and 240 feet wide. That's the dimensions of the field. So if we wanted to do it in square feet, we would multiply these two numbers. So how many square feet is it? Well, I'll do it out here. I'll do it on the left. So let me multiply it out. We have 240 times 363. Actually, I want to do it with a little bit more space. So we're going to do it in blue. Maybe I should do it here. I don't think it will confuse you. So I have 240 times 363. 3 times 0 is 0. 3 times 4 is 12. Carry the 1. 3 times 2 is 6, plus 1 is 7. So it's 720. We can ignore that for a little bit. Throw a 0 down here. 6 times 0 is 0. 6 times 4 is 24. 6 times 2 is 12, plus this 2 is 14. And then we have our last row to deal with. So we're going to add two 0's because we're dealing with the hundreds place. 3 times 0 is 0. 3 times 4 is 12. This was 4, carry the 1. 3 times 2 is 6, plus 1 is 7. And now we can just add everything up. 0. 2. 7 plus 4 is 11. Carry the 1. 1 plus 4 is 5. 5 plus 2 is 7. And then we have 1 plus 7 is 8. So the field is 87,120 square feet. Now, they want to know how many acres is the field? And they say an acre is 43,560. And you see all your choices are whole numbers. And you can just eyeball this, that 87,000 whatever is roughly twice 43,560. Right? 80,000 is roughly twice 40,000. If you wanted to just hand-wave it, you could just say, well, that's just choice number A, or that's just 2, right? This looks like 2 times that. And if you wanted to verify it, 2 times-- let's verify it. Let me do it in a color that you can actually see. So 43,560-- you wouldn't actually have to do this on the test, but I want to show you that our approximation works. 2 times 0 is 0. 2 times 6 is 12. 2 times 5 is 10, plus the 1 is 11. 2 times 3 is 6, plus 1 is 7. 2 times 4 is 8. And we've got the exact number which was the square footage of the field. So it's 2 acres. So if you do it like that, it's 2 acres. 1 acre and 2 acres, just like that. 37. The object below is made of 10 rectangular prisms-- that just means these three-dimensional rectangular things-- each with dimensions of 5 centimeters by 3 centimeters by 2 centimeters. What is the volume, in cubic centimeters, of the object? So it's kind of this plus sign, or this cross looking thing. And they do this-- I mean, these things are really just to throw you off. So what is the volume of each of these little cubes right there? So its volume is 5 times 2 times 3. So each cube has a volume of, what's 5 times 2? Is 10, times 3 is 30. So each has 30 cubic centimeters. Do it in a darker color. Each of these is 30 cubic centimeters. And how many of these are there in this structure? Well, we have 2 layers. The top layer has 1 box here, that's 1, 2-- this is the second box-- 2, 3, 4, and then 5. So the top layer has 5 in this plus sign. I could draw it like this. 1, 2, 3, 4, 5. So if we have 5 in the top layer, and we have 2 layers. So we have 10 of these boxes, right? So 10 boxes, or rectangular prisms if you want to call it that, times 30 cubic centimeters per box. The boxes cancel out, you get 10 times 30 cubic centimeters, and you get 300 cubic centimeters is the volume of this entire structure. And that is choice D. I literally think they drew these little diagonals just to confuse you, just to maybe make you think that you have to do something with triangles. But no, you just have to count the number of boxes and figure out the volume of each of them. We had 10 boxes, each of them has a volume of 30 cubic centimeters, so 300 cubic centimeters total.