Determine the vertex form and the maximum or minimum value of the function.
f(x) = 2xsecond power+ 8x + 3
To determine the vertex form of the function, we need to complete the square:
f(x) = 2x^2 + 8x + 3
f(x) = 2(x^2 + 4x) + 3
f(x) = 2(x^2 + 4x + 4 - 4) + 3
f(x) = 2((x + 2)^2 - 4) + 3
f(x) = 2(x + 2)^2 - 5
Therefore, the vertex form of the function is f(x) = 2(x + 2)^2 - 5, with vertex at (-2, -5).
To determine the maximum or minimum value of the function, we can look at the coefficient of the squared term. In this case, since the coefficient is positive, the parabola opens upwards and the vertex represents the minimum value of the function. Therefore, the minimum value of the function is f(-2) = 2(-2 + 2)^2 - 5 = -5.
One week, a construction worker bought 40 1/10 pounds of nails. The next week, he bought 2 1/2 times as many nails as the week before. How many pounds of nails did he buy the second week?
To solve the problem, we first need to convert 40 1/10 pounds of nails into a fraction. We do this by multiplying the whole number (40) by the denominator of the fraction (10) and adding the numerator (1), then putting the result over the denominator:
40 1/10 = (40 x 10 + 1)/10 = 401/10
So the worker bought 401/10 pounds of nails in the first week.
To find out how many pounds of nails he bought in the second week, we need to multiply the amount from the first week (401/10) by 2 1/2. To multiply by a mixed number, we first need to convert it to an improper fraction:
2 1/2 = (2 x 2 + 1)/2 = 5/2
Now we can multiply:
401/10 x 5/2
To simplify this fraction, we can first cancel out a factor of 2 in the numerator and denominator of the second fraction:
401/10 x 5/2 = 401/10 x 5/2 x 2/2 = 401/10 x 5/1 = 2005/10
Finally, we can convert this improper fraction back to a mixed number by dividing the numerator by the denominator:
2005/10 = 200 5/10 = 200 1/2
Therefore, the worker bought 200 1/2 pounds of nails in the second week.
Evaluate: 5-3|3x(-6)|
First, we need to evaluate the expression inside the absolute value brackets before we can multiply by -3.
-3|3x(-6)| = -3|-18|
Next, we need to evaluate the absolute value of -18, which is just the positive value of 18:
-3|-18| = -3(18)
Now we can multiply by -3 to get our final answer:
-3(18) = -54
Therefore, 5-3|3x(-6)| = -54.
Jermaine earned $2000 during the summer. He plans to put the money in a savings account and leave it for 40 years. The account will earn 9% simple interest annually. What will the balance be after 40 years?
To find the balance after 40 years, we can use the simple interest formula:
I = P * r * t
where I is the interest earned, P is the principal (the initial amount invested), r is the annual interest rate as a decimal, and t is the time in years.
In this case, we know that Jermaine's initial investment (the principal) is $2000, the annual interest rate is 9%, and the time is 40 years. We can substitute these values into the formula, remembering to convert the interest rate from a percentage to a decimal:
I = 2000 * 0.09 * 40
I = 7200
So the interest earned over 40 years is $7200. To find the final balance, we add the interest earned to the initial investment:
Final balance = $2000 + $7200 = $9200
Therefore, after 40 years, the account balance will be $9200.
Quadrilateral JKLM is a kite with an area of 63 square feet. What is the length of NM?
In a kite, the two diagonals are perpendicular and intersect at their midpoint. Let's call the intersection point of JK and LM as point O. Then, we can split the kite into two congruent triangles, JOK and LOM, and find their individual areas, which we can add together to get the total area of the kite.
Let's call NM as x. Then, we can use the area formula for a triangle:
Area = (1/2) * base * height
For triangle JOK, the base is JN, which is half of NM, and the height is OK. For triangle LOM, the base is ML, which is also half of NM, and the height is OL. So we can write:
Area of JOK = (1/2) * (1/2)x * OK = (1/4)xOK
Area of LOM = (1/2) * (1/2)x * OL = (1/4)xOL
Since the two triangles are congruent, their areas are equal, so we can add them together and set them equal to the area of the kite:
(1/4)xOK + (1/4)xOL = 63
Simplifying this equation, we get:
(1/4)x(OK + OL) = 63
But we know that OK and OL are just the same distance from point O, which is the midpoint of JK and LM. So OK = OL, and we can substitute 2OK for OK + OL:
(1/4)x(2OK) = 63
Multiplying both sides by 4/2, we get:
(1/2)xOK = 126
xOK = 252
Therefore, the length of OK (and also OL) is 252. But we still need to find NM, which is twice OK:
NM = 2 * OK = 2 * 252 = 504
So the length of NM is 504 feet.
Julia and Brady are staying in the same hotel. Julia takes Elevator 1 and stops 40 feet above the hotel lobby. Brady takes Elevator 2 and stops 24 feet below the hotel lobby.
Which expression can be used to calculate how far apart Julia and Brady are?
To find how far apart Julia and Brady are, we need to calculate the total distance between their two elevator stops, which is the sum of the distance above the lobby that Julia stops and the distance below the lobby that Brady stops. We can express this as:
Distance apart = 40 + (-24)
Note that we use a negative sign for the distance below the lobby, since it is a downward distance. Simplifying this expression, we get:
Distance apart = 16
Therefore, Julia and Brady are 16 feet apart.
719 milliliters =liters
To convert milliliters to liters, we divide by 1000. So:
719 milliliters = 719/1000 liters
Simplifying the fraction, we get:
719/1000 = 0.719
Therefore, 719 milliliters is equal to 0.719 liters.
-x/6+29=-30
To solve for x, we want to isolate the variable on one side of the equals sign.
Subtracting 29 from both sides first:
-x/6 + 29 - 29 = -30 - 29
Simplifying:
-x/6 = -59
Multiplying both sides by -6 to eliminate the fraction:
(-6)*-x/6 = -59*(-6)
x = 354
Therefore, x = 354.
The Friday attendance at a local theater is shown for the last 7 weeks. What is the median Friday attendance for the 7-week period?
To find the median attendance, we need to arrange the attendance numbers in order from smallest to largest. Then, we can find the middle number. If there are an odd number of attendance numbers, the median is simply the middle number. If there are an even number of attendance numbers, the median is the average of the two middle numbers.
Arranging the attendance numbers from smallest to largest, we get:
87, 90, 92, 94, 95, 96, 97
Since there are seven attendance numbers, which is an odd number, the median is simply the middle number, which is 94.
Therefore, the median Friday attendance for the 7-week period is 94.