A candy factory wants to compare a new machine to their current machine. Both machines produce y boxes of candy for every hour x. The current machine is represented by the equation f(x). The new machine is represented by the equation g(x).
f(x) = 15x + 90
g(x) = 30x + 30
After what number of hours will the amount of candy produced be the same for both machines?
To find the number of hours when the amount of candy produced is the same for both machines, we need to set the two equations equal to each other and solve for x:
15x + 90 = 30x + 30
Subtract 15x from both sides:
90 = 15x + 30
Subtract 30 from both sides:
60 = 15x
Divide both sides by 15:
x = 4
After 4 hours, the amount of candy produced will be the same for both machines.
consider the following angle, measured in radians: x=7π/6
what is the measure of the angle in degrees
To convert the angle from radians to degrees, we can use the conversion factor:
1 radian = 180/π degrees.
Given that x = 7π/6, we can substitute that value into the conversion factor:
x degrees = (7π / 6) * (180 / π)
Simplifying the expression:
x degrees = (7 * 180) / 6
x degrees = 1260 / 6
x degrees = 210
Therefore, the measure of the angle x = 7π/6 radians is 210 degrees.
What expression is equal to 8 + 2(y + 5)?
The expression equal to 8 + 2(y + 5) can be simplified using the distributive property. We can distribute the 2 to both terms inside the parentheses:
8 + 2(y + 5) = 8 + 2y + 2(5)
Simplifying further:
8 + 2(y + 5) = 8 + 2y + 10
Finally, combining like terms:
8 + 2(y + 5) = 18 + 2y
Therefore, the expression equal to 8 + 2(y + 5) is 18 + 2y.
Which of the following is an arithmetic sequence?
a) f(x)=x+y
b) f(x)=12x^2
c) f(x)=3x+y
d) f(x)=3x+12
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. To determine which of the given functions represent an arithmetic sequence, we need to find the common difference between terms.
a) f(x) = x + y: This function does not have a constant difference, as the second term is y units greater than the first term.
b) f(x) = 12x^2: This function represents a quadratic equation and does not have a constant difference between terms.
c) f(x) = 3x + y: Similarly, this function does not have a constant difference between terms.
d) f(x) = 3x + 12: This function represents a linear equation in the form y = mx + b. Here, the coefficient of x (3) represents the constant difference between terms.
Therefore, option d) f(x) = 3x + 12 represents an arithmetic sequence.
The volume of a sphere is two-thirds the volume of a cylinder.
What is the volume of a cylinder that has the same diameter as a sphere with a volume of 14 u3, in u3?
The volume of a sphere is given by the formula V_s = (4/3)πr^3, where r is the radius of the sphere.
Let's assume that the diameter of the sphere is also the diameter of the cylinder. In that case, the radius of the cylinder is half the radius of the sphere.
The volume of the cylinder is given by the formula V_c = πr_c^2h, where r_c is the radius of the cylinder and h is the height of the cylinder.
Since the diameter of the sphere is the same as the diameter of the cylinder, the radius of the cylinder is half the radius of the sphere. Therefore, r_c = r/2.
We are given that the volume of the sphere is two-thirds the volume of the cylinder, so we can write the equation as:
(4/3)πr^3 = (2/3)π(r/2)^2h
Simplifying:
(4/3)πr^3 = (2/3)π(r^2/4)h
Multiplying both sides by 3/π:
4r^3 = (r^2/4)h
Dividing both sides by r^2:
4r = h/4
Multiplying both sides by 4:
16r = h
Now we can use the volume of the sphere to find the radius:
(4/3)πr^3 = 14
Multiplying both sides by 3/(4π):
r^3 = 42/(4π)
Taking the cube root of both sides:
r = (42/(4π))^(1/3)
Now we can substitute the value of r into the equation we found for the height:
16r = h
16 * (42/(4π))^(1/3) = h
Calculating the approximate value of h using a calculator, we get:
h ≈ 9.912 u^3
Therefore, the volume of the cylinder is approximately 14π cubic units (since the height is an approximation).
The volume of a sphere is two-thirds the volume of a cylinder.
What is the volume of a cylinder that has the same diameter as a sphere with a volume of 14 u^3, in u^3?