Type the correct answer in the box. Write your answer as a whole number. The radius of the base of a cylinder is 10 centimeters,
and its height is 20 centimeters. A cone is used to fill the cylinder with water. The radius of the cone's base is 5 centimeters, and its height is 10 centimeters. The number of times one needs to use the completely filled cone to completely fill the cylinder with water is ______.
Well, we would find the volume of both the cylinder and the cone. The formula to find the volume of a cylinder is V = \pi r^{2} h. The formula to find the volume of a cone is V = \frac{1}{3} \pi r^{2} h. Now, when using pi (the symbol \pi ), if the problem asks to use the approximate value of pi as 3.14, we'll use 3.14 to represent pi. If the problem asks for the exact answer, then we'll just put the pi sign next to our final answer we've gotten. We can do both ways to solve with pi. Tho let's solve using the approximate value of pi. One more thing. When there's an exponent in the problem, (such as r^{2} in the problem), that just means you would multiply the base, (which is r in r^{2} ), by itself the number of times the exponent, (which is the number 2 in r^{2} ), shows. For example, if it was 3^{2} , that would just mean we would multiply 3 times itself twice. So 3^{2} would be the same as 3 times 3, which is 9. When you deal with variables in a problem, (the letters used to represent a value in the equation, such as a, b, c, d, etc.), they're solved in multiple ways. If a number is next to a variable, (for example, 3b), it would mean you are supposed to multiply the number times the variable. If a number is next to parenthesis, that would mean you would multiply the number times the answer from the parenthesis. In formulas, when all the letters and numbers are squished together in one line, that means you would multiply all of them times each other after they're individually solved. In this problem, r = radius and h = height. So without further-a-do, let's begin! :) So the question says that the radius of the cylinder is 10 centimeters, and the height is 20 centimeters. According to the formula given above, we would multiply the radius times itself twice, which would be 10 times 10, which equals 100, then multiply it by the height which is 20, so 100 times 20 = 2,000. If we were to find the exact volume, it would be 2,000 with the pi sign next to it, which would be 2,000 \pi . Though, let's find the volume with the approximate value of pi, 3.14. So 2,000 times 3.14 is 6280. The exact volume of the cylinder is 2,000 \pi and the approximate volume is 6280. Now after we find the volume of the cone, we would need to find out how many times we'd need to use the cone to fill up the cylinder's volume. To find the volume of the cone, we would do the same as last time. Since the cone's radius is 5 centimeters, and its height is 10 centimeters, we would first multiply the radius times itself twice, which would be 5 times 5, which is 25, then multiply that by the height, which would be 25 times 10, which is 250. The fraction part of the formula means this- the numerator "1" means you would multiply what you have so far times 1, and the denominator "3" means you will divide what you have so far NOW by 3. So 250 times 1 = 250, and 250 divided by 3 = 83.3333333333, though we'd say that answer up to the hundredths place, which would be 83.33. If you need the exact answer, we'd put the pi sign next to it as 83.33 \pi , and you're done.Tho we didn't find the approximate volume of the cone.So we'd trace back to what answer we had before moving on to the step with the fraction \frac{1}{3} . Since we were on 250, we'd multiply that by 3.14, which is 785, and continue with what we did with the fraction. 785 times 1 = 785, and 785 divided by 3 = 261.666666667, and saying the answer up to the hundredths place, the approximate volume of the cone would be 261.66 Now since we know the volume of the cylinder, exact = 2,000 \pi and approximate volume = 6280, this means this is how much of the cylinder should be filled when we use the cone to pour in the water. We could easily determine this by division. Finding with the exact volume, you would do 2,000 (from cylinder's volume) divided by 83.33 (from cone's volume), which is 24.0009600384, and since it's not a whole answer, you would move it up a whole number. Why you may ask? Well you can't pour 0.0009600384 of a cone's volume into the cylinder now can you? :P So it would take 25 times of pouring the cone filled with water into the cylinder in order for the cylinder to be full using the exact volumes of both objects. Now finding with the approximate volumes of both objects, we'd do the same we did last time. The cylinder's volume divided by the cone's volume, which is 6280 divided by 261.66, would be 24.0006114805, and saying the answer up to the hundredths place, 24.00, but for the same reason as the last one when we were using the exact volumes, we'd round 24 up a whole number, so it would approximately take 25 times to fill the cylinder with the cone. Either way, using exact or approximate volumes, your final answer would be 25 times. =DI hope I helped! ^-^
At first piecewise functions may be strange confusing things, but they aren't so bad. I like to think of it like this: f(x) is a function that changes its identity based on what the input x is. We have three situations
f(x) = -4x+3 when x < 3
f(x) = -x^3 when
f(x) = 3x^2+1 when x > 8
In a sense, we have three different functions but they are combined somehow.
If x is smaller than 3, then we go for the first definition. Or if x is between 3 and 8, then we go for the second definition. Or if x is larger than 8, then we go for the third definition.
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f(-5) means f(x) when x = -5. We see that -5 is smaller than 3, so x = -5 makes x < 3 true. We'll use the first definition
f(x) = -4x+3
f(-5) = -4(-5)+3
f(-5) = 20+3
f(-5) = 23
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Now the input is x = 12. This is larger than 8. In other words, x = 12 makes x > 8 true. We'll use the third definition
f(x) = 3x^2+1
f(12) = 3(12)^2+1
f(12) = 3(144)+1
f(12) = 432+1
f(12) = 433
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Side notes:
We won't use the second definition since we don't have any x inputs between 3 and 8
To say "less than or equal to" on a keyboard, you can write "<=" without quotes. For example, is the same as x<=5
is the same as x<=5