Answers:
Problem #1: Option C, 166 yards cubed
Problem #2: Option A, 10
Step-by-step solution:
The first problem may seem a lot more confusing that it really is and this actually happened to me when I looked at it. Just because the cylinder is slanted that actually doesn't change any of the volume that we would have if it was straight up with the same diameter and height.
Using what we know about cylinders, let us plug all the information inside of the formula to determine the volume of the cylinder. However, before doing that we need to determine the radius of the circle from the diameter that was given.
Divide the diameter by two to get the radius
- [tex]Radius = \frac{Diameter}{2}[/tex]
- [tex]Radius = \frac{5.2\ yd}{2}[/tex]
- [tex]Radius = 2.6\ yd[/tex]
Plug in the values
- [tex]V_{cylinder} = (\pi * radius^2)*height[/tex]
- [tex]V_{cylinder} = (\pi * (2.6\ yd)^2)*7.8\ yd[/tex]
Simplify the exponent
- [tex]V_{cylinder} = (\pi * (2.6)^2 *(yd)^2)*7.8\ yd[/tex]
- [tex]V_{cylinder} = (\pi * 6.76\ yd^2)*7.8\ yd[/tex]
Simplify the expression
- [tex]V_{cylinder} = (21.237\ yd^2)*7.8\ yd[/tex]
- [tex]V_{cylinder} = 165.65\ yd^3[/tex]
Therefore, after simplifying the expression using the cylinder volume formula and the given information we were able to determine that the option that best fits the description of our answer is option C, 166 cube yards.
Now that we completed the first question, we can move onto the second question. In this question we are given a figure along with 3 numbers and one unknown, x, which we need to find the value of. We need to use the Secant-Secant Power Theorem to help determine our solution.
Make an expression to represent the scenario
- [tex]32(x + 32) = 28(20 + 28)[/tex]
We know have an expression which we can now simplify and get the value of the unknown easily. The first step would be to distribute the 32 and 28.
Distribute both sides
- [tex]32(x + 32) = 28(20 + 28)[/tex]
- [tex](32 * x) + (32 * 32) = (28 * 20) + (28 * 28)[/tex]
- [tex]32x + 1024 = 560 + 784[/tex]
Simplify the expression
- [tex]32x + 1024 = 1344[/tex]
We now have a simple expression where we can now subtract both sides by 1024 to help isolate x with its coefficient.
Subtract 1024 from both sides
- [tex]32x + 1024 - 1024 = 1344 - 1024[/tex]
- [tex]32x = 1344 - 1024[/tex]
The final step that we have to help fully isolate x by itself is to divide both sides by 32 which would remove the coefficient from x.
Divide both sides by 32
- [tex]\frac{32x}{32} = \frac{320}{32}[/tex]
- [tex]x = \frac{320}{32}[/tex]
After simplifying the expression completely, we are able to see that the best option that fits our description is option A, 10.
