A toy baseball bat comes with 3 plastic balls. The balls are in a box that is a rectangular solid. The box is just big enough to
hold the 3 balls. The radius of the balls is 1.4 in. each What is the volume of the air in the box surrounding the balls?
1 answer:
Answer:
31.4 in³
Step-by-step explanation:
The box is just big enough to hold the 3 balls, so it must have a length 6 times the radius of each ball, a width 2 times the radius, and a height 2 times the radius.
The volume of the box is:
V = (6r)(2r)(2r)
V = 24r³
The volume of the 3 balls is:
V = 3 (4/3 π r³)
V = 4πr³
So the volume of the air is:
V = 24r³ − 4πr³
V = (24 − 4π) r³
Since r = 1.4 inches:
V = (24 − 4π) (1.4 in)³
V ≈ 31.4 in³
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I made my own graph using the given points. Pls. see attachment.
<span>Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.
I used paint program to graph the points based on the given data.
I used a ruler as guide as I draw my line containing both points D and F and extended the line outward.
The yellow lines above represents the shaded portion of the graph.
y = mx + b
b is the y-intercept. It is the value of y when x is 0. In this case, b = 3.5
m is the slope. It is the change in y divided by the change in x.
point D (2,4) ; point F (-2,3)
change in y: 4 - 3 = 1
change in x: 2 - (-2) = 4
m = 1/4
y = 1/4 x + 3.5
Since we are looking for a system of inequality and the portion shaded is in the upper part of the line, the value of y will be greater than or equal to.
y </span>≥ 1/4x + 3.5<span>
Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A.
To check the solution given above. We will use the x values of points D and F and check whether its y values are equal to or greater than the solution.</span>
y ≥ 1/4x + 3.5
Point D (2,4) : x = 2 : y ≥ 1/4 * 2 + 3.5 → 0.5 + 3.5 → 4
Point F (-2.3): x = -2 ; y ≥ 1/4 * -2 + 3.5 → -0.5 + 3.5 → 3
<span>
Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised.
y > 2x - 2
We will get the coordinates of each point and see if its x value will generate the desired value of y.
A(2,-3) ; x = 2 : </span><span>y > 2(2) - 2 </span>→ 4 - 2 → 2 . DID NOT PASS. -3 is lesser than 2.<span>
B(-3,-4) ; x = -3 : y > 2(-3) - 2 </span>→ -6 - 2 → -8. PASSED. -4 is greater than -8.<span>
C(-4,2) ; x = -4 : y > 2(-4) - 2 </span>→ -8 - 2 → -10. PASSED. 2 is greater than -10.<span>
D(2,4) ; x = 2 : y > 2(2) - 2 </span>→ 4 - 2 → 2. PASSED. 4 is greater than 2.<span>
E(3,1) ; x = 3 : y > 2(3) - 2 </span>→ 6 - 2 → 4. DID NOT PASS. 1 is lesser than 4.<span>
F(-2,3) ; x = -2 ; y > 2(-2) - 2 </span>→ -4 - 2 → -6. PASSED. 3 is greater than -6.
Among the farms, points B,C, D, and F can raise chickens based on the given system of linear inequality.
<span>Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.
I used paint program to graph the points based on the given data.
I used a ruler as guide as I draw my line containing both points D and F and extended the line outward.
The yellow lines above represents the shaded portion of the graph.
y = mx + b
b is the y-intercept. It is the value of y when x is 0. In this case, b = 3.5
m is the slope. It is the change in y divided by the change in x.
point D (2,4) ; point F (-2,3)
change in y: 4 - 3 = 1
change in x: 2 - (-2) = 4
m = 1/4
y = 1/4 x + 3.5
Since we are looking for a system of inequality and the portion shaded is in the upper part of the line, the value of y will be greater than or equal to.
y </span>≥ 1/4x + 3.5<span>
Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A.
To check the solution given above. We will use the x values of points D and F and check whether its y values are equal to or greater than the solution.</span>
y ≥ 1/4x + 3.5
Point D (2,4) : x = 2 : y ≥ 1/4 * 2 + 3.5 → 0.5 + 3.5 → 4
Point F (-2.3): x = -2 ; y ≥ 1/4 * -2 + 3.5 → -0.5 + 3.5 → 3
<span>
Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised.
y > 2x - 2
We will get the coordinates of each point and see if its x value will generate the desired value of y.
A(2,-3) ; x = 2 : </span><span>y > 2(2) - 2 </span>→ 4 - 2 → 2 . DID NOT PASS. -3 is lesser than 2.<span>
B(-3,-4) ; x = -3 : y > 2(-3) - 2 </span>→ -6 - 2 → -8. PASSED. -4 is greater than -8.<span>
C(-4,2) ; x = -4 : y > 2(-4) - 2 </span>→ -8 - 2 → -10. PASSED. 2 is greater than -10.<span>
D(2,4) ; x = 2 : y > 2(2) - 2 </span>→ 4 - 2 → 2. PASSED. 4 is greater than 2.<span>
E(3,1) ; x = 3 : y > 2(3) - 2 </span>→ 6 - 2 → 4. DID NOT PASS. 1 is lesser than 4.<span>
F(-2,3) ; x = -2 ; y > 2(-2) - 2 </span>→ -4 - 2 → -6. PASSED. 3 is greater than -6.
Among the farms, points B,C, D, and F can raise chickens based on the given system of linear inequality.