Solve.
The quotient 4 over x plus the quotient 12 over x squared equals 5
A. x equals six fifths or x equals 2
B. x equals negative six fifths or x equals negative 2
C. x equals six fifths or x equals negative 2
D. x equals negative six fifths or x equals 2
We can start by combining the two quotients into a single fraction with a common denominator:
4/x + 12/x^2 = (4x + 12) / x^2
Setting this equal to 5, we can cross-multiply to get:
4x + 12 = 5x^2
Rearranging and setting the equation equal to zero, we have:
5x^2 - 4x - 12 = 0
This quadratic equation can be factored as:
(5x + 18)(x - 2) = 0
So the solutions are:
5x + 18 = 0 OR x - 2 = 0
x = -18/5 OR x = 2
Therefore, the answer is (D) x equals negative six fifths or x equals 2.
Patty and Mike are installing new floors in their house. Working alone, Mike can complete the floor in 10 hours. Patty can complete the same floor in 8 hours if working alone. How long will it take them, working together, to finish the floor? Round your answer to the nearest hundredth if necessary.
A. 9.00 hours
B. 0.23 hours
C. 4.44 hours
D. 18.00 hours
Let's use the formula:
1 / time taken = work rate
Let Mike's work rate be m and Patty's work rate be p. Then we can write:
m = 1/10 (because he can do 1 floor in 10 hours)
p = 1/8 (because she can do 1 floor in 8 hours)
When they work together, their work rates add up:
m + p = 1/10 + 1/8
To combine these fractions, we need a common denominator:
m + p = 4/40 + 5/40
m + p = 9/40
So, working together, their combined work rate is 9/40. Now we can use the formula again:
1 / time taken = work rate
1 / t = 9/40
Solving for t, we get:
t = 40/9 ≈ 4.44
Therefore, it will take them approximately 4.44 hours to finish the floor together. The answer is (C).
The time t required to drive a certain distance varies inversely with the speed r. If it takes 4 hours to drive the distance at 35 miles per hour, how long will it take to drive the same distance at 45 miles per hour?
A. about 3.11 hours
B. 140 hours
C. about 5.14 hours
D. 393.75 hours
We know that the time t varies inversely with the speed r, which means:
t = k/r
where k is a constant of proportionality. We can solve for k using the given information that it takes 4 hours to drive the distance at 35 miles per hour:
4 = k/35
Multiplying both sides by 35, we get:
k = 140
Now we can use this value of k to find the time it would take to drive the same distance at 45 miles per hour:
t = 140/45
t ≈ 3.11
Therefore, it would take about 3.11 hours to drive the same distance at 45 miles per hour. The answer is (A).
Do the data in the table represent a direct variation or an inverse variation? Write an equation to model the data in the table.
x 1 3 4 7
y 5 15 20 35
A. direct variation; y equals one fifth times x
B. inverse variation; x times y equals 5
C. direct variation; y equals 5 times x
D. inverse variation; x times y equals one fifth
Looking at the table, we can see that as x increases, y increases as well. This suggests a direct variation, where y is proportional to x. To confirm this, we can check if the ratios y/x are all equal:
5/1 = 15/3 = 20/4 = 35/7
Since the ratios are equal, we have a direct variation. To find the equation that models this data, we can use the formula y = kx, where k is the constant of proportionality. To solve for k, we can use any of the pairs of x and y values:
5 = k(1) or k = 5/1
So the equation that models the data is:
y = 5x
Therefore, the answer is (C) direct variation; y equals 5 times x.
What are the excluded values of the function? y equals negative 2 over the quantity 5 times x plus 40
A. x = 0
B. x equals negative 6
C. x equals negative 40
D. x equals negative 8
The fraction in the function has a denominator of (5x + 40), which means this denominator cannot be zero. Therefore, we need to find the value of x that makes the denominator zero to determine the excluded value.
5x + 40 = 0
5x = -40
x = -8
So the excluded value of the function is x = -8, since it would make the denominator zero. The answer is (D) x equals negative 8.
Which of these would be the most energy efficient for you to consume according to what you know about trophic levels?(1 point)
Responses
tuna
tuna
hamburger
hamburger
omelet
omelet
applesauce