The polynomial -x^3-x^2+12x represents the volume of a rectangular aquatic tank in Cubic feet. The length of the tank is (x+4).
A. Use synthetic formula to help factor the volume of the polynomial. How many linear factors should you look for? B. What are the dimensions of the tank? C.Find the value of x that will maximize the volume of the box. D. What is the maximum volume?
1 answer:
Answer:
A. x(x +4)(3 -x) . . . . a cubic will have 3 linear factors
B. The dimensions are x, x+4, 3 -x, or 1.694 ft, 5.694 ft, 1.306 ft
C. x = (√37 -1)/3 ≈ 1.694254 ft
D. Maximum volume: ≈ 12.597 ft³
Step-by-step explanation:
A. The first attachment shows the synthetic division of the given polynomial by x+4. The result is -x^2 +3x = x(3 -x). Hence the factored polynomial is ...
x(3-x)(x+4) . . . . . . shown as positive
Since the polynomial is a cubic, we know there will be 3 roots. For this problem, that means we expect 3 factors.
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B. We are apparently to assume that the dimensions of the tank correspond to the three factors:
- x
- 3-x
- x+4
For the maximum size tank (see part C), the corresponding measurements in feet are
- 1.694 ft
- 1.306 ft
- 5.694 ft
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C. Tank volume will be maximized when the derivative of volume with respect to x is zero. The derivative polynomial is ...
dV/dx = -3x^2 -2x +12
Solving this by the usual methods, we find the positive value of x to be ...
x = (-1 +√37)/3 ≈ 1.694254 . . . . feet
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D. Evaluating the polynomial for this value of x gives ...
volume ≈ 12.5972 . . . cubic feet
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Answer:
-8
Step-by-step explanation:
Let's first establish the reference line (the one that the second line will be perpendicular to). We are told that this line passes through two points:
(1,1) and (-3,-1).
We'll find a line equation using the point slope form format to start:
(y - y1) = m * (x - x1), where m is the slope and the x and y are from two points.
(x,y) = (1,1)
(x1,y1) = (-3,-1)
Rearrange the equation:
(y - y1) = m * (x - x1)
m = (y - y1)/ (x - x1)
m = (1-(-1))/(1-(-3))
m = 2/4, or 1/2: The slope is 1/2. [<u>This is "m."]</u>
We can use the slope-intercept form for this line (y=mx + b) and then calculate b, the y-intercept:
y = (1/2)x + b
Use either of the two given points. I'll use (1,1) since I have memorized the "1" math tables.
y = (1/2)x + b
1 = (1/2)(1) + b for (1,1)
b = 1/2
This makes the reference line: y = (1/2)x+(1/2)
===
The line perpendicular must have a slope that is the negative inverse of (1/2). This would be -(2/1), or -2.
We can then write y = -2x + b
To find be, enter the one given point for this line: (-3,-2)
y = -2x + b
-2 = -2(-3) + b
-2 = 6 + b
b = -8
The perpendicular line is thus:
y = -2x - 8
It has a y-intercept of -8
Answer:
A = $3,926.71
Step-by-step explanation:
Given: Principal (P) = $3200, Annual Rate (R) = 4.1%, Time = 5 years
To find: How much money would he have in the account after 5 years, if he made no deposits or withdrawals during that time?
Formula:
Solution: Compound interest is one of the most important concepts to understand when managing your finances. It can help you earn a higher return on your savings and investments, but it can also work against you when you're paying interest on a loan
First, convert R as a percent to r as a decimal
r = R/100
r = 4.1/100
r = 0.041 rate per year,
Then solve the equation for A
A = P(1 + r/n)
A = 3,200.00(1 + 0.041/12)
A = 3,200.00(1 + 0.003416667)
A = $3,926.71
Hence, Jay would have $3,926.71 after 5 years is if he made no deposits or withdrawals during that time.