The lengths of the sides of a rectangle are a and b, where a > b The sum of the lengths of the two shorter sides and one of the longer sides of the rectangle is 36. What value of a maximizes the area of the rectangle?
A .9
B. 12
C. 18
D. 24
The answer is C. I suspect there is an easier way to solve than completing the square and finding the vertex of the resulting quadratic function. What is your most direct, easy to understand solution to this calculator-allowed question?
g(x) = 1/5 (5)^(x+4)
For the given function g, which of the following equivalent forms shows the y-coordinate of the y-intercept of the graph of y=g(x) in the xy-plane as a constant or coefficient?
A. g(x) = 125(5)^x
B. g(x) = 25(5)^(x+1)
C. g(x) = 5(5)^(x+2)
D. g(x) = (5)^(x+3)
Choice A is correct, but why?
x f(x)
____________
-5 0
-1 36
0 0
1 6
2 0
The function f is defined by a polynomial. Some values of x and f(x) are shown in the table above. Which of the following could define f?
A) (x – 5) (x + 2)
B) (x + 5)^2 (x – 2)^3
C) x^2 (x + 5) (x – 2)^2
D) x (x + 5) (x – 2)
Specifically, can you explain why C is correct but D cannot be?
In a circle with area of 120 to 124 sq inches, the area of the sector formed by an angle is between 20 and 21 sq inches. What is one possible integer value of the angle?
I came up with a low value of 60 (if area of circle is on the lowest end, 120 sq inches, and area of the sector is also on the low end, 20 inches). If both those areas are on the highest end, then I came up with 60 again. But answer is supposed to be 59 ≤ x ≤ 63. Does this make sense, and if so, can you explain it?
Can you explain a more direct way to solve College Board Official Practice Test 9, Math Section 4 #19, than the College Board’s explanation? I seem to remember something about making a chart to solve mixture problems. Would that work here?
A question on polynomials practice question #10 (p. 149 in PWN the SAT Math Guide): I understand how to solve using polynomial long division but can you explain your shortcut from the answer explanation? How would we know to plug in -3?
Could you further explain your explanation of Practice Question #4 in PWN the SAT Math Guide “Polynomials” chapter (p. 148). In your long division confirmation of the answer (explanation p. 326), how do you know to divide by 9x + 1?
Can you suggest a quick method to solve College Board Practice Test 5 Math Section 3 #14? I plugged in easy numbers for x in order to eliminate answer choices (first x=1, then x=0), but wonder if there is a more direct way to solve.
In the “Proving Grounds” Quiz 1 #5, this question is given:
p(x) = ax^2 + bx + c
q(x) = ax^2 + bx + d
The functions p(x) and q(x) are defined above, and a, b, c, and d are constants. If x – 3 is a factor of p(x) and d is 8 greater than c, what is the remainder when q(x) is divided by x – 3?
In your answer explanation, you state:
“…We know that, as a general rule, the remainder when a polynomial f(x) is divided by x – a is f(a).”
Can you explain this further?
Is there a way to solve this system of equations without using the quadratic formula (or graphing)?
f(x) = –2/3 x + 4
g(x) = 3(x + 2)^2 – 4
How many solutions does the system above have?
y > -½ x + a
y > 2 x – a
If the value of a is 7, which of the following is a possible solution to the system of inequalities above?
A) (2, 6)
B) (4, 5)
C) (5, 4)
D) (5, 3)
Is there a better way than plugging in the answers to solve this?
f(x) = √x
g(x) = 3x – b
If the graph of y = f(g(x)) passes through (6, 5) in the standard (x, y) coordinate plane, what is the value of b?
a = 4800 – 6t
b = 5400 – 8t
In the system of equations above, a and b represent the distance, in meters, two marathon runners are from the finish line after running for four hours and t seconds. How far will runner a be from the finish line when runner b passes her?
A. 300 meters
B. 500 meters
C. 100 meters
D. 3000 meters
A rideshare app charges $2 per trip plus $0.4 per mile. A competitor charges $1 for the first 6 miles plus $0.5 per mile for every additional mile. For what length trip would the two services charge the same amount?
A. 10 miles
B. 18 miles
C. 20 miles
D. 40 miles
Can you craft an algebraic equation to solve this directly – or is plugging in the answers the way to go?
This may be a little advanced for the SAT, but complex numbers sometimes show up –as do cubic polynomials– so hopefully you can address this for me! TIA!
Which of the following could be the full set of complex roots of a cubic polynomial with real coefficients?
A. { 0, 1, i}
B. {1, i, 2i}
C. {2, i}
D. {3, 2 + i, 2 – i}
